HSA.SSE.B.3: Equivalent Forms of Quadratic and Exponential Expressions
In plain English: HSA.SSE.B.3 is the Common Core algebra standard that asks students to choose and produce an equivalent form of an expression to reveal and explain properties of the quantity it represents. Students factor a quadratic to show its zeros, complete the square to show its maximum or minimum value, and use exponent properties to rewrite exponential expressions. It is usually taught in Algebra I and revisited in Algebra II.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a.Factor a quadratic expression to reveal the zeros of the function it defines.
b.Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c.Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Common Core State Standards for Mathematics · Domain: Seeing Structure in Expressions (SSE) · Cluster: Write expressions in equivalent forms to solve problems Also written as HSA-SSE.B.3 or A-SSE.3 · Official standard
One quadratic or exponential function can be written in several equivalent forms, and each form makes a different property easy to read. In this lesson students factor a quadratic to reveal its zeros, complete the square to reveal its maximum or minimum value, and use the properties of exponents to rewrite an exponential expression so that it shows a growth or decay rate for a different time unit.
The emphasis is on choosing a form for a purpose. Students should leave able to answer two questions for any expression: which form shows the property I need, and what does that property mean for the quantity the expression represents?
Learning Objectives
By the end of this lesson, students will be able to:
Factor a quadratic expression and use the factored form to state the zeros of the function it defines
Complete the square to write a quadratic in vertex form and state its maximum or minimum value and where it occurs
Use the properties of exponents to rewrite an exponential expression so that it shows a rate for a different time unit
Choose the equivalent form that reveals a requested property and explain what that property means in context
Prior Knowledge Required
Students should already be comfortable with:
Using the structure of an expression to rewrite it, including recognizing a difference of squares HSA.SSE.A.2
Applying the properties of integer exponents, including (bm)n = bmn8.EE.A.1
Multiplying binomials and combining like terms
Knowing that a zero of a function is an input where the output is 0
Write three expressions on the board without saying that they are equal:
Warm-Up Prompt
"Here are three expressions: x2 - 2x - 8, (x + 2)(x - 4), and (x - 1)2 - 9. Evaluate each one at x = 0, x = 4 and x = 1. What do you notice? Which expression made each calculation easiest?"
Students should find that all three give -8, 0 and -9, so the expressions are equivalent. Ask which one makes it obvious that the output is 0 at x = 4 (the factored form) and which makes it obvious that the output can never be less than -9 (the vertex form, because a square is never negative). Record the idea "same function, different form, different information" at the top of the board.
Direct Instruction25 minutes
Present one decision routine, then model it with five examples that cover all three parts of the standard:
Name the property you need: zeros, a maximum or minimum, or a rate per time unit.
Pick the form that shows it: factored form for zeros, vertex form a(x - h)2 + k for the extreme value, an exponent rewritten with the new time unit for a rate.
Rewrite and check equivalence: expand, or substitute one input into both forms.
Read the property and interpret it: state what the number means for the quantity, including whether it makes sense.
Factor to reveal zeros (part a)
"A ball is tossed from a 48-foot platform. Its height in feet after t seconds is h(t) = -16t2 + 32t + 48. When does it hit the ground?"
Equation: -16t2 + 32t + 48 = -16(t - 3)(t + 1), zeros t = 3 and t = -1, so it lands after 3 seconds (t = -1 is outside the context)
Complete the square (part b)
"Find the minimum value of f(x) = x2 - 6x + 11."
Equation: x2 - 6x + 9 + 2 = (x - 3)2 + 2, minimum value 2 at x = 3
Complete the square in context (part b)
"A club sells 200 - 5p T-shirts when the price is p dollars, so revenue is R(p) = -5p2 + 200p. What price gives the most revenue?"
Equation: -5(p2 - 40p + 400) + 2000 = -5(p - 20)2 + 2000, maximum $2,000 at p = $20
Exponent properties, new time unit (part c)
"An investment grows as 1000(1.15)t, where t is in years. About what is the monthly growth rate?"
Equation: 1000(1.15)t = 1000(1.151/12)12t ≈ 1000(1.0117)12t, about 1.17% per month
Exponent properties, exact rewrite (part c)
"A culture of 500 cells doubles every 3 hours: N(t) = 500 · 2t/3, t in hours. By what factor does it grow each day?"
Equation: 500 · 2t/3 = 500 · (28)t/24 = 500 · 256t/24, so it multiplies by 256 each day
For Example 4, stop and ask why the monthly rate is not 15% ÷ 12 = 1.25%. Growth compounds: 1.011712 is about 1.15, while 1.012512 is about 1.161. This is a common error on part c.
Guided Practice15 minutes
Pairs work through three short tasks, one per part, with a class debrief after each. (1) Factor x2 + 2x - 24 and state where the graph crosses the x-axis (x = -6 and x = 4). (2) Complete the square for x2 + 4x + 1 and state the minimum (-3 at x = -2). (3) Rewrite 80(0.64)t, t in years, to show the half-year decay factor: 0.64t = (0.8)2t, so the quantity loses 20% every half year. Circulate and listen for students who find the vertex but report the x-value as the minimum.
Independent Practice15 minutes
Students complete four problems alone: one factoring problem with a leading coefficient other than 1, one completing-the-square problem with a leading coefficient other than 1, one context problem that asks for a maximum, and one exponential rewrite from a yearly to a quarterly rate. For each problem, students must write one sentence naming the property the new form reveals.
Closure5-10 minutes
Exit ticket: "For f(x) = x2 - 10x + 16, write the form you would use to find the zeros and the form you would use to find the minimum. Give both forms and both answers." (Answers: (x - 2)(x - 8), zeros 2 and 8; (x - 5)2 - 9, minimum -9 at x = 5.)
Differentiation Strategies
For Struggling Students
Give a factoring table: list factor pairs of the constant term and their sums before students write any binomials
Use algebra tiles or an area model to show why x2 - 6x needs 9 more to become the square (x - 3)2
For part c, start with exact rewrites such as 8t = 23t before moving to decimal bases and rounding
For Advanced Students
Complete the square for ax2 + bx + c in general and show that the extreme value occurs at x = -b/(2a)
Compare 1.15t rewritten monthly and weekly, and explain why the per-period rates are not proportional to the period length
Find a quadratic that cannot be factored over the integers and explain what completing the square still tells you about its zeros
Assessment Guidance
What to Look For
Look for students who can state which form they chose and why, not only students who can carry out the algebra. Common errors to watch: sign errors when reading zeros from factored form (x + 2 gives the zero -2, not 2), reporting the x-coordinate of the vertex as the maximum or minimum value, forgetting to factor out the leading coefficient before completing the square, and dividing an annual percent rate by 12 instead of taking the twelfth root of the growth factor.
02
Classroom Activities
3 Activities
1
Form Match-Up
20 minGroups of 3-4
Each group gets 18 cards: six functions in standard form, and for each one a factored or vertex form card and a property card (such as "zeros at -3 and 5" or "minimum value 2"). Groups build sets of three and explain which form made each property visible.
Setup
Quadratic sets, for example: x2 - 2x - 15 with (x - 5)(x + 3) and "zeros 5 and -3"; x2 + 8x + 18 with (x + 4)2 + 2 and "minimum 2 at x = -4"
Exponential sets, for example: 9t with 32t and "triples every half unit of time"; 1.21t with 1.12t and "grows 10% every half year"
Add two distractor cards with sign errors, such as (x + 5)(x - 3) for x2 - 2x - 15
Procedure
Groups have 12 minutes to form the sets and check each match by expanding or by substituting a value
Each group explains one distractor to the class: what error would produce it, and how substitution exposes it
Modification for Distance Learning
Put the cards on a shared slide or whiteboard app and have groups drag them into columns in breakout rooms.
2
Best Price Challenge
25 minPairs
Pairs receive a revenue or profit model for a small business and must complete the square to find the best price, then check the answer with a table of values. The goal is to connect the vertex form to a decision someone would actually make.
Scenarios
Food truck: selling 300 - 20p tacos at p dollars gives R(p) = -20p2 + 300p = -20(p - 7.5)2 + 1125, so the best price is $7.50 for $1,125
Car wash: selling 120 - 4p washes at p dollars gives R(p) = -4p2 + 120p = -4(p - 15)2 + 900, so the best price is $15 for $900
Concert: selling 1200 - 30p tickets at p dollars gives R(p) = -30p2 + 1200p = -30(p - 20)2 + 12000, so the best price is $20 for $12,000
Discussion Questions
What do the numbers h and k in -a(p - h)2 + k mean for the business?
Factor each revenue function. What do its zeros mean, and why is the best price exactly halfway between them?
Why is the maximum revenue not the same as the maximum number of items sold?
3
Rate Translator
20 minIndividual then share
Students rewrite exponential models so that they report a rate for a different time unit, then compare their answer with a partner who used a calculator table. The activity targets part c of the standard and the compounding error.
Given Models
A town of 12,000 people grows as 12000(1.04)t, t in years: find the growth rate per decade (1.0410 ≈ 1.480, about 48% per decade)
A medicine dose of 400 mg decays as 400(0.5)t/6, t in hours: find the hourly decay rate (0.51/6 ≈ 0.891, about 10.9% per hour)
A savings account grows as 2000(1.06)t, t in years: find the quarterly growth rate (1.061/4 ≈ 1.0147, about 1.47% per quarter)
Share and Check
Partners check each rewrite by raising the new factor back to the right power, for example 1.01474 ≈ 1.06
Each student writes one sentence explaining why 48% per decade is more than 10 × 4%
Poster Variation
Assign each student a model from science or finance and have them make a small poster showing the original expression, the exponent rule used, the rewritten expression and a one-sentence interpretation.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Three Equivalent Forms of f(x) = x² - 2x - 8
The same parabola written three ways. The factored form (x + 2)(x - 4) gives the zeros -2 and 4, the vertex form (x - 1)2 - 9 gives the minimum value -9 at x = 1, and the standard form gives the y-intercept -8. The graph is drawn to scale; the dashed line x = 1 is the axis of symmetry.
Diagram 2: Rewriting Exponential Expressions for a New Time Unit
The power of a power rule lets you change the time unit without changing the function. In the first row the rewritten base is rounded, so the new form is approximately equal; in the second row the rewrite is exact because 28 = 256.
04
Homework Assignment
~30 min
HSA.SSE.B.3 Homework: Choosing the Right Form
Directions: For each problem, (a) write the requested equivalent form, (b) show that it is equivalent by expanding or by checking one input value, (c) state the property the form reveals, and (d) when there is a context, write one sentence explaining what the property means.
Part 1: Factor to Reveal Zeros (Problems 1-2)
Factor f(x) = x2 + x - 12 and state the zeros of f.
Factor g(x) = 2x2 - 7x - 15 and state the zeros of g. Check one zero by substituting it into the original expression.
Part 2: Complete the Square to Reveal a Maximum or Minimum (Problems 3-4)
Complete the square to write p(x) = x2 + 8x + 7 in the form (x - h)2 + k. State the minimum value of p and the x-value where it occurs.
A model rocket's height in feet after t seconds is h(t) = -16t2 + 96t + 4. Complete the square to find the maximum height and the time when it is reached.
Part 3: Use Exponent Properties (Problems 5-6)
An account balance is B(t) = 3500(1.05)t, where t is in years. Rewrite B so the base shows the growth factor per quarter (round the base to four decimal places). What is the approximate quarterly growth rate?
The amount of a medicine in a patient's body is M(t) = 250(0.5)t/4 milligrams, where t is in hours. (a) Rewrite M to show the hourly decay factor (round to three decimal places) and state the hourly percent decrease. (b) Rewrite M to show the daily decay factor exactly.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Equivalent Form
Correct form, all steps shown
Correct method, one algebra or sign error
Form missing or not equivalent
Equivalence Check
Expansion or substitution confirms the form
Check attempted but incomplete
No check
Property Stated
Zeros, extreme value or rate stated correctly
Property stated with a sign or rounding error
Property missing or misidentified
Interpretation
Clear sentence with units and context
Sentence present but vague or missing units
No interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Each expression below is being compared with f(x) = x2 - x - 20. Which one is equivalent to f(x) and shows the zeros of f directly?
Answer: C
(x - 5)(x + 4) expands to x2 - x - 20, and a product is 0 when a factor is 0, so the zeros are 5 and -4. Choices A and B are also equivalent to f(x), but neither shows the zeros: B is the vertex form and shows the minimum -81/4. Choice D has the signs reversed; it expands to x2 + x - 20.
Question 2 of 20 · Multiple Choice
What are the zeros of f(x) = (2x - 3)(x + 6)?
Answer: A
Set each factor equal to 0: 2x - 3 = 0 gives x = 3/2, and x + 6 = 0 gives x = -6. Choice B reverses both signs. Choice C forgets to divide by the coefficient 2, and choice D divides the wrong way.
Question 3 of 20 · Multiple Choice
Which statement about f(x) = x2 + 10x + 21 is correct?
Answer: D
The factors of 21 that add to 10 are 3 and 7, so f(x) = (x + 3)(x + 7), and the zeros are the inputs that make each factor 0: -3 and -7. Choice A has the right factors but reads the zeros with the wrong sign. Choice C multiplies to 21 but its middle term would be 22x.
Question 4 of 20 · Multiple Choice
A ball's height in feet after t seconds is h(t) = -16t2 + 48t + 64 = -16(t - 4)(t + 1). When does the ball hit the ground?
Answer: B
The ball is on the ground when h(t) = 0, which happens at t = 4 and t = -1. Time after the throw cannot be negative, so the ball lands after 4 seconds. Choice C lists both zeros without rejecting the one that does not fit the context. Choice D uses the starting height, not a zero.
Question 5 of 20 · Multiple Choice
Which expression is equivalent to x2 - 8x + 19 and is written in vertex form?
Answer: A
Half of -8 is -4, and (-4)2 = 16, so x2 - 8x + 19 = (x2 - 8x + 16) + 3 = (x - 4)2 + 3. Choice B has the wrong sign inside the square. Choice C adds 16 without subtracting it, and choice D uses the whole coefficient -8 instead of half of it.
Question 6 of 20 · Multiple Choice
What is the minimum value of f(x) = x2 + 10x + 18?
Answer: C
Completing the square gives f(x) = (x2 + 10x + 25) - 7 = (x + 5)2 - 7. The square is never negative, so the smallest output is -7, reached at x = -5. Choice A is the x-value where the minimum occurs, not the minimum value. Choice B is the y-intercept.
Question 7 of 20 · Multiple Choice
Which statement is true for g(x) = -(x - 5)2 + 12?
Answer: B
Because -(x - 5)2 is never positive, g(x) is at most 12, and it equals 12 when x = 5. The negative sign makes it a maximum, not a minimum (choice A). Choice C swaps the roles of h and k, and choice D reads the sign of h incorrectly.
Question 8 of 20 · Multiple Choice
A theater sells 400 - 10p tickets when the price is p dollars, so its revenue is R(p) = -10p2 + 400p. Completing the square gives R(p) = -10(p - 20)2 + 4000. What is the maximum revenue?
Answer: D
In vertex form the maximum value is k = 4000, so the most revenue is $4,000, at a price of $20 (200 tickets). Choice A is the best price, not the revenue. Choice B is the number of tickets sold at a price of $0.
Question 9 of 20 · Multiple Choice
A population is modeled by P(t) = 800(1.44)t, where t is in years. Which equivalent expression shows the growth factor for each half year?
Answer: B
Since 1.22 = 1.44, we have 1.44t = (1.22)t = 1.22t, and 2t counts half years. The population grows 20% every half year. Choice A halves the rate, but 1.222 = 1.4884, not 1.44. Choice C is not equivalent, and choice D changes the starting value.
Question 10 of 20 · Multiple Choice
Which expression is equivalent to 52t?
Answer: C
By the power of a power rule, 52t = (52)t = 25t. Choice A multiplies the base by 2 instead of squaring it. Choice B equals 2 · 5t, which is different. Choice D squares the base and keeps the 2 in the exponent, so the 2 is used twice.
Question 11 of 20 · Multiple Choice
An investment grows as 1.08t, where t is in years. Which expression is approximately equivalent and shows the monthly growth factor?
Answer: A
1.08t = (1.081/12)12t, and 1.081/12 ≈ 1.0064, so the growth is about 0.64% per month. Choice B divides the rate by 12 (8% ÷ 12 ≈ 0.67%), which ignores compounding: 1.006712 ≈ 1.083, not 1.08. Choice C multiplies the exponent without changing the base.
Question 12 of 20 · Multiple Choice
Which expression is equivalent to 3t + 2?
Answer: D
By the product rule, 3t + 2 = 3t · 32 = 9 · 3t. This shows that the function starts at 9 when t = 0. Choice A adds instead of multiplying. Choice B multiplies 3 by 2 instead of raising 3 to the second power. Choice C multiplies the exponents instead of adding them.
Question 13 of 20 · Multiple Choice
A student needs the minimum value of f(x) = x2 - 4x - 12. Which equivalent form shows it directly?
Answer: B
x2 - 4x - 12 = (x2 - 4x + 4) - 16 = (x - 2)2 - 16, so the minimum is -16 at x = 2. Choice A is equivalent but shows the zeros 6 and -2, not the minimum. Choice D is not equivalent: it adds 4 to the constant instead of subtracting it.
Question 14 of 20 · Multiple Choice
A savings account grows as A(t) = 1500(1.03)4t, where t is in years. What is the approximate annual growth rate?
Answer: C
(1.03)4t = (1.034)t ≈ 1.1255t, so the balance grows about 12.55% per year. Choice A adds four quarters of 3% and ignores compounding. Choice B is the rate per quarter, not per year.
Question 15 of 20 · Short Answer
Factor f(x) = 3x2 - 12 completely and state the zeros of f.
f(x) = 3(x - 2)(x + 2). First factor out the common factor 3, then use the difference of squares x2 - 4 = (x - 2)(x + 2). The zeros are x = 2 and x = -2. Factoring out 3 does not change the zeros, because 3 is never 0.
Question 16 of 20 · Short Answer
Factor g(x) = x2 - 9x + 14 and state the zeros of g.
Find two numbers that multiply to 14 and add to -9: -2 and -7. So g(x) = (x - 2)(x - 7), and the zeros are x = 2 and x = 7. Check: g(2) = 4 - 18 + 14 = 0.
Question 17 of 20 · Short Answer
Complete the square to write f(x) = 2x2 - 12x + 23 in vertex form. State the minimum value and where it occurs.
Factor 2 out of the x-terms: 2(x2 - 6x) + 23. Add and subtract 9 inside: 2(x2 - 6x + 9) - 18 + 23. So f(x) = 2(x - 3)2 + 5, and the minimum value is 5 at x = 3. A common error is subtracting 9 instead of 2 · 9 = 18.
Question 18 of 20 · Short Answer
A rectangular pen has a perimeter of 40 meters. Its area is A(w) = w(20 - w), where w is the width in meters. Complete the square to find the largest possible area and the width that gives it.
A(w) = -w2 + 20w = -(w2 - 20w + 100) + 100 = -(w - 10)2 + 100. The maximum area is 100 square meters when w = 10 m, so the pen is a 10 m by 10 m square. The width must be between 0 and 20 m for the pen to exist.
Question 19 of 20 · Short Answer
A bacteria population is N(t) = 300 · 4t, where t is in hours. Rewrite N(t) to show how the population changes every half hour.
Since 4 = 22, 4t = (22)t = 22t, so N(t) = 300 · 22t. The exponent 2t counts half hours, so the population doubles every half hour. The rewrite is exact.
Question 20 of 20 · Short Answer
A car's value is V(t) = 24000(0.85)t dollars, where t is in years. Rewrite V(t) to show the monthly decay factor (round to four decimal places), and explain why the monthly percent decrease is not 15% ÷ 12.
0.85t = (0.851/12)12t ≈ 0.986512t, so V(t) ≈ 24000(0.9865)12t, a decrease of about 1.35% per month. Dividing 15% by 12 gives 1.25%, but each month's decrease is taken from a smaller value, so the monthly factors multiply: 0.986512 ≈ 0.85, while 0.987512 ≈ 0.86.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
Why rewrite an expression in an equivalent form?
Because each equivalent form makes a different property of the quantity easy to see, and HSA.SSE.B.3 asks students to choose a form on purpose for that reason. The three sub-standards name the cases: factor a quadratic to show its zeros (a), complete the square to show its maximum or minimum value (b), and use exponent properties to transform an exponential expression (c). The key word is "choose": students should know which form to reach for, not only how to produce it.
Why does factored form show the zeros?
A product equals 0 only when one of its factors is 0. In f(x) = (x - 5)(x + 3), the output is 0 exactly when x - 5 = 0 or x + 3 = 0, so the zeros are 5 and -3. Standard form x2 - 2x - 15 hides this information, because a sum of terms has no such property.
Why does vertex form show the maximum or minimum?
In f(x) = a(x - h)2 + k, the square (x - h)2 is never negative and equals 0 only at x = h. If a > 0, the smallest output is k (a minimum). If a < 0, the largest output is k (a maximum). Students can reason this out without memorizing a rule, which is what the standard means by "explain."
What mistakes do students often make on this standard?
Reading zeros with the wrong sign: (x + 4) gives the zero -4
Reporting the x-coordinate of the vertex as the maximum or minimum value
Completing the square without factoring out the leading coefficient, or subtracting the wrong amount afterward
Treating 32t as 6t, or 2t + 3 as 2t + 8
Dividing an annual percent rate by 12 to get a monthly rate, which ignores compounding
Does every quadratic have to be factorable for this standard?
No. Part a expects students to factor quadratics that can be factored, usually over the integers or rationals. When a quadratic such as x2 + 6x + 2 does not factor nicely, completing the square still works: (x + 3)2 - 7 shows the minimum, and it also leads to the zeros -3 ± √7. Solving quadratic equations in general belongs to HSA.REI.B.4.
How is part c different from just knowing the exponent rules?
Students learned the rules with numbers in grade 8. Part c asks them to use the rules on expressions that model something, and to say what the new form means. For example, 1.21t = 1.12t is a rule application, but the payoff is the interpretation: the quantity grows 10% every half year. Ask for that sentence every time.
Is a rounded rewrite like 1.15^t ≈ 1.0117^(12t) still an equivalent expression?
The exact rewrite (1.151/12)12t is equivalent. Once the base is rounded to 1.0117, the new expression is only approximately equal, and students should write ≈ instead of =. It is worth showing that the error is small by computing 1.011712 ≈ 1.1500.
How does HSA.SSE.B.3 show up on the SAT?
The digital SAT's Advanced Math domain includes questions about equivalent expressions and about choosing a form of a quadratic or exponential function that displays a feature such as a zero, a vertex or a growth factor. The skills in this lesson are directly relevant there.
How does HSA.SSE.B.3 connect to HSF.IF.C.8?
They are companion standards. HSA.SSE.B.3 focuses on producing the equivalent expression. HSF.IF.C.8 applies the same moves to functions and adds interpretation of features such as symmetry of the graph and percent rates of change in context. Many teachers teach them together, and this lesson already asks for interpretation.
What comes after HSA.SSE.B.3?
HSA.REI.B.4: solving quadratic equations by factoring, completing the square and the quadratic formula
HSF.IF.C.8: using equivalent forms to interpret functions in context
HSA.APR.B.3: using zeros from factored form to sketch graphs of higher-degree polynomials
HSN.RN.A.2: working with rational exponents such as 1.151/12
07
Related Standards
6 standards
These standards connect to HSA.SSE.B.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.EE.A.1Prerequisite
Apply properties of integer exponents to generate equivalent numerical expressions
