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HSA.REI.B.4Common CoreMathAlgebraGrades 9-12

HSA.REI.B.4: Solving Quadratic Equations in One Variable

In plain English: HSA.REI.B.4 is the Common Core algebra standard that asks students to solve quadratic equations in one variable by inspection, square roots, factoring, completing the square or the quadratic formula, as fits the equation. Students also derive the quadratic formula by completing the square and write complex solutions as a ± bi. Real solutions are usually taught in Algebra I and complex ones in Algebra II.

Solve quadratic equations in one variable.

  1. a.Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
  2. b.Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Common Core State Standards for Mathematics · Domain: Reasoning with Equations and Inequalities (REI) · Cluster: Solve equations and inequalities in one variable
Also written as HSA-REI.B.4 or A-REI.4 · Official standard
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01

Lesson Plan

70-80 min

Overview

Students learn to solve quadratic equations in one variable by choosing the method that fits the equation's initial form: inspection, taking square roots, factoring, completing the square, or the quadratic formula. The lesson starts with completing the square, because it turns any quadratic equation into the form (x - p)² = q with the same solutions, and that form is exactly what students use to derive the quadratic formula.

Students then use the formula on equations whose discriminant b² - 4ac is negative, recognize that the solutions are complex, and write them in the form a ± bi. Throughout, students check their answers by substitution and, in context problems, decide which solutions make sense.

Learning Objectives

By the end of this lesson, students will be able to:

  • Complete the square to rewrite any quadratic equation in x as an equation of the form (x - p)² = q with the same solutions
  • Derive the quadratic formula by completing the square on ax² + bx + c = 0
  • Choose an efficient method (inspection, square roots, factoring, completing the square, or the quadratic formula) based on the initial form of the equation
  • Recognize from the discriminant when the quadratic formula gives complex solutions and write them as a ± bi
  • Check solutions by substitution and interpret them in context

Prior Knowledge Required

Students should already be comfortable with:

  • Square roots of positive numbers and solving x² = p 8.EE.A.2
  • Solving linear equations in one variable HSA.REI.B.3
  • Factoring quadratic expressions and the zero product property HSA.SSE.A.2
  • Simplifying radicals such as √40 = 2√10
  • The imaginary unit i, where i² = -1 HSN.CN.A.1, for the complex-solutions part of the lesson

Lesson Procedure

70-80 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Post three equations and ask students to solve them without writing any algebra steps:

    Warm-Up Prompt

    "Find every number that makes each equation true: (1) x² = 81, (2) (x - 2)² = 81, (3) x² = -81. For which one can you not find a real number? Why?"

    Collect answers. Many students give 9 for the first equation; push for -9 as well. For (2), students should see that x - 2 must be 9 or -9, so x = 11 or x = -7. For (3), no real number squares to a negative number. Leave that question open and tell students they will return to it at the end of the lesson. Point out that equations (1) and (2) were easy because the squared quantity was already alone on one side: that is the form the lesson builds toward.

  2. Direct Instruction25 minutes

    Part 1: Completing the square (standard a). Use Diagram 1 to show why x² + 6x needs 9 more to become a perfect square. Then show the general steps to transform a quadratic equation into the form (x - p)² = q:

    1. Make the leading coefficient 1: if the equation is ax² + bx + c = 0 with a ≠ 1, divide every term by a.
    2. Move the constant: rewrite as x² + (b/a)x = -c/a.
    3. Add the square of half the x-coefficient to both sides: here that is (b/(2a))² = b²/(4a²).
    4. Write the left side as a square: (x + b/(2a))² = (b² - 4ac)/(4a²). This is (x - p)² = q with p = -b/(2a) and q = (b² - 4ac)/(4a²).
    5. Take square roots and solve: x + b/(2a) = ±√(b² - 4ac)/(2a), so x = (-b ± √(b² - 4ac))/(2a).

    Do the derivation twice side by side: once with a numerical equation such as 2x² + 6x + 1 = 0 and once with letters. Stress that every step produces an equation with the same solutions as the one before, so the formula gives exactly the solutions of the original equation. When taking square roots in step 5, note that √(4a²) = 2|a|, and the ± sign covers both signs of a.

    Part 2: Choosing a method (standard b). Work through the examples below. Before each, ask: "What does the initial form of this equation suggest?"

    • Completing the square

      Rewrite x² + 6x - 7 = 0 in the form (x - p)² = q and solve.

      Equation: (x + 3)² = 16, so x = 1 or x = -7

    • Taking square roots

      The squared expression is already isolated after dividing by 3: 3(x - 2)² = 75.

      Equation: (x - 2)² = 25, so x = 7 or x = -3

    • Factoring

      The trinomial factors over the integers: 2x² + 5x - 3 = 0.

      Equation: (2x - 1)(x + 3) = 0, so x = 1/2 or x = -3

    • Quadratic formula, irrational solutions

      No integer factors: 2x² - 4x - 3 = 0 with a = 2, b = -4, c = -3.

      Equation: x = (4 ± √40)/4 = 1 ± √10/2

    • Quadratic formula, complex solutions

      x² - 4x + 13 = 0 has discriminant 16 - 52 = -36.

      Equation: x = (4 ± 6i)/2 = 2 ± 3i

    For the last example, return to the warm-up question x² = -81. With i² = -1, the solutions are x = ±9i. Explain that whenever b² - 4ac < 0, the formula produces the square root of a negative number, and the solutions are a pair of complex numbers with real part -b/(2a) and imaginary parts ±√(4ac - b²)/(2a). Name the letter clash out loud: the a and b in a ± bi are not the coefficients a and b of the equation. Use Diagram 2 to connect the sign of the discriminant to the number of real solutions.

  3. Guided Practice15-20 minutes

    Pairs solve four equations, one at a time, and must name their method before starting: x² - 50 = 0 (square roots: ±5√2), x² + 8x + 15 = 0 (factoring: -3, -5), x² - 10x + 20 = 0 (completing the square: 5 ± √5), and 4x² + 4x + 5 = 0 (formula: -1/2 ± i). After each one, ask a pair that used a different method to share. Listen for these errors: dropping the negative root when taking square roots, adding (b/2)² to only one side, forgetting to divide by a before completing the square, and dividing only part of the numerator by 2a.

  4. Independent Practice15 minutes

    Students complete six problems on their own: x² = 64 (inspection: ±8), 2(x + 1)² = 32 (square roots: 3, -5), x² + x - 12 = 0 (factoring: 3, -4), x² - 4x - 3 = 0 rewritten as (x - 2)² = 7 (2 ± √7), 3x² + 2x - 2 = 0 (formula, irrational: (-1 ± √7)/3), and x² + 2x + 10 = 0 (formula, complex: -1 ± 3i). For each, students write the method in the margin and check one solution by substitution. For the complex case, students check by substituting a + bi and using i² = -1.

  5. Closure5-10 minutes

    Exit ticket: (1) Rewrite x² - 2x - 5 = 0 in the form (x - p)² = q. (Answer: (x - 1)² = 6.) (2) Without solving, decide whether 3x² + x + 2 = 0 has real or complex solutions, and explain how you know. (Discriminant 1 - 24 = -23, so complex.) (3) Name the method you would use first for 5x² = 45, and why.

Differentiation Strategies

For Struggling Students

  • Use algebra tiles or grid paper to build the square for x² + bx before completing the square symbolically, starting with even values of b
  • Give a method-choice checklist: Is the squared expression isolated? Does it factor? If not, use the formula
  • Have students write a, b, and c in a small table before substituting into the formula, including the signs

For Advanced Students

  • Ask students to complete the square on ax² + bx + c = 0 without dividing by a first (multiply by 4a instead) and compare the result with the usual derivation
  • Ask for a quadratic equation with integer coefficients whose solutions are 3 ± 2i, and explain how the sum and product of the solutions relate to the coefficients
  • Ask students to prove that if the coefficients are real and the discriminant is negative, the two solutions are always complex conjugates

Assessment Guidance

What to Look For

Check that students can explain why each step of completing the square keeps the same solutions, not only carry out the steps. When a student uses the formula on an equation that factors or is already in square-root form, ask them to name a faster method: standard b expects the method to fit the initial form. For complex solutions, look for the final answer written as a ± bi with the real part and imaginary part simplified separately, such as 1/2 ± (3/2)i rather than (2 ± 6i)/4.

02

Classroom Activities

3 Activities

1

Build the Square with Tiles

20 minPairs

Students use algebra tiles (or squares and strips drawn on grid paper) to see why completing the square adds (b/2)², then connect each physical step to a line of algebra that ends in the form (x - p)² = q.

Procedure

  • Give each pair an x² tile and eight x tiles. Ask them to arrange x² + 8x as close to a square as possible by splitting the x tiles evenly on two sides
  • Ask: how many unit tiles fill the missing corner? (16.) Record x² + 8x + 16 = (x + 4)²
  • Now solve x² + 8x - 9 = 0: move the 9, add 16 to both sides, and write (x + 4)² = 25, so x = 1 or x = -9
  • Repeat with x² + 10x + 21 = 0 ((x + 5)² = 4, so x = -3 or x = -7) and x² - 6x + 2 = 0 ((x - 3)² = 7, so x = 3 ± √7)

Discussion Questions

  • Why do we add the same number to both sides, not only to the side with the tiles?
  • What happens with x² + 5x? Can you still complete the square? (Yes: add 25/4.)
  • In (x - p)² = q, what does it mean for the solutions if q is negative?

Modification for Distance Learning

Use a free online algebra tiles tool or a shared slide with draggable squares and strips. Pairs share their screen to show the finished square before writing the equation.

2

Method Match Relay

20 minGroups of 3-4

Groups receive 12 equation cards. For each card, the group must agree on the most efficient method for its initial form, solve it, and check one solution. The goal is to build judgment about method choice, which is the second half of standard b.

Sample Cards

  • Inspection: x² = 144 (±12), x² = 0 (0)
  • Square roots: (x + 1)² = 49 (6, -8), 2x² - 18 = 0 (±3), (x - 4)² = 12 (4 ± 2√3)
  • Factoring: x² - x - 20 = 0 (5, -4), 3x² + 7x + 2 = 0 (-1/3, -2), x² - 9x = 0 (0, 9)
  • Completing the square or formula: x² + 4x - 1 = 0 (-2 ± √5), 2x² - 3x - 4 = 0 ((3 ± √41)/4)
  • Complex: x² + 2x + 5 = 0 (-1 ± 2i), x² + 16 = 0 (±4i)

Procedure

  • One student per round reads a card and proposes a method; the next student solves; the third checks by substitution; roles rotate
  • Groups sort finished cards into five piles by method and tape them to a poster
  • Each group presents one card where members disagreed about the method and explains their final choice

Challenge Variation

Give groups blank cards and ask them to write one equation for each method, where that method is clearly the best choice. Groups trade cards and solve each other's equations.

3

Derive the Formula Side by Side

25 minPairs

Students derive the quadratic formula in two columns: the left column completes the square on a numerical equation, and the right column does the same step with a, b, and c. This makes standard a concrete: the general derivation is the same process students already used with numbers.

Setup

  • Left column equation: 2x² + 6x + 1 = 0. Right column equation: ax² + bx + c = 0 with a ≠ 0
  • Provide a handout with five blank rows labeled: divide by a, move the constant, add the square of half the x-coefficient, write as a square, take square roots and solve

Procedure

  • Partner A completes a row on the left; Partner B completes the matching row on the right; they swap roles on the next row
  • Left column should reach (x + 3/2)² = 7/4 and x = (-3 ± √7)/2
  • Right column should reach (x + b/(2a))² = (b² - 4ac)/(4a²) and x = (-b ± √(b² - 4ac))/(2a)
  • Pairs check the left-column answer by substituting a = 2, b = 6, c = 1 into the finished formula

Discussion Questions

  • Which row in the right column is the form (x - p)² = q? What are p and q?
  • Why must a be nonzero?
  • Where in the derivation can you already tell whether the solutions will be real or complex?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: Completing the Square as an Area Model

Start: x² + 6x x² 3x 3x 9 missing x 3 x 3 Full square: (x + 3)² = x² + 6x + 9 Solve x² + 6x - 7 = 0 x² + 6x = 7 x² + 6x + 9 = 7 + 9 (x + 3)² = 16 x + 3 = 4 or x + 3 = -4 x = 1 or x = -7 Form (x - p)² = q with p = -3, q = 16 Add (half of 6)² = 9 to both sides.
Splitting 6x into two strips of 3x leaves a 3-by-3 corner missing. Adding 9 to both sides completes the square and turns x² + 6x - 7 = 0 into (x + 3)² = 16, an equation with the same solutions. The strips for 3 are drawn at a fixed scale; the length x is unknown.

Diagram 2: The Discriminant and the Number of Real Solutions

x² - 4x + 3 = 0 1 2 3 4 2 4 b² - 4ac = 4 > 0 Two real solutions: x = 1, 3 x² - 4x + 4 = 0 1 2 3 4 2 4 b² - 4ac = 0 One real solution: x = 2 x² - 4x + 5 = 0 1 2 3 4 2 4 b² - 4ac = -4 < 0 No x-intercepts: x = 2 ± i
Graphs of y = x² - 4x + c for c = 3, 4, 5, drawn to scale. Each real solution of the equation is an x-intercept. When b² - 4ac < 0 the parabola does not cross the x-axis, and the quadratic formula gives two complex solutions, here 2 ± i.

04

Homework Assignment

~30 min

HSA.REI.B.4 Homework: Solving Quadratic Equations

Directions: Show all work. For each equation in Part 2, name the method you chose and explain in one sentence why it fits the equation's form. Check at least one solution of every equation by substitution. Write complex solutions in the form a ± bi.

Part 1: Completing the Square and the Formula (Problems 1-2)

  1. Rewrite x² - 10x + 18 = 0 in the form (x - p)² = q. State p and q, then solve the equation.
  2. Complete the square on ax² + bx + c = 0 (a ≠ 0) to show that it becomes (x + b/(2a))² = (b² - 4ac)/(4a²). Take square roots of both sides of that form to derive the quadratic formula. Then use your result to find p and q for 2x² + 8x - 3 = 0 and solve that equation.

Part 2: Choosing a Method (Problems 3-4)

  1. Solve each equation using the method that best fits its form: (a) x² = 121 (b) (x + 4)² = 18 (c) x² + 2x - 35 = 0 (d) 3x² - 5x - 1 = 0
  2. A rectangular garden is 3 meters longer than it is wide, and its area is 70 square meters. Write a quadratic equation for the width w, solve it, and explain which solution makes sense. Give the garden's dimensions.

Part 3: Complex Solutions (Problems 5-6)

  1. Use the quadratic formula to solve (a) x² + 6x + 13 = 0 and (b) 2x² - 6x + 5 = 0. Write each solution as a ± bi.
  2. A ball is thrown upward, and its height in feet after t seconds is h = -16t² + 32t + 5. Set up an equation to find when the ball is 25 feet high and solve it. What does your answer tell you about the ball? Find its maximum height to confirm.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Completing the SquareCorrect (x - p)² = q form with p and q statedCorrect method with one arithmetic errorForm missing or incorrect
Method ChoiceEfficient method named and justified for each equationMethods correct but not justifiedNo method named
AccuracyAll solutions correct, both roots given, checkedMost solutions correct or a root missingMost solutions incorrect
Complex Solutions and ContextWritten as a ± bi and interpreted correctlyCorrect values, form or interpretation incompleteMissing or incorrect

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

  1. Question 1 of 20 · Multiple Choice

    Solve by inspection: x² = 169.

  2. Question 2 of 20 · Multiple Choice

    What number must be added to x² + 14x to make a perfect square trinomial?

  3. Question 3 of 20 · Multiple Choice

    Which equation is x² - 8x + 3 = 0 rewritten in the form (x - p)² = q?

  4. Question 4 of 20 · Multiple Choice

    Solve (x - 5)² = 36.

  5. Question 5 of 20 · Multiple Choice

    What are the solutions of x² - 3x - 28 = 0?

  6. Question 6 of 20 · Multiple Choice

    Use the quadratic formula to solve x² + 4x + 1 = 0.

  7. Question 7 of 20 · Multiple Choice

    What does the discriminant tell you about the solutions of 3x² - 2x + 4 = 0?

  8. Question 8 of 20 · Multiple Choice

    Solve x² + 25 = 0.

  9. Question 9 of 20 · Multiple Choice

    Solve x² - 4x + 29 = 0 and write the solutions in a ± bi form.

  10. Question 10 of 20 · Multiple Choice

    Which method best fits the initial form of 4(x + 1)² = 100?

  11. Question 11 of 20 · Multiple Choice

    In deriving the quadratic formula, you reach x² + (b/a)x = -c/a. What do you add to both sides next?

  12. Question 12 of 20 · Multiple Choice

    A student solves x² = 6x by dividing both sides by x and gets x = 6. What is wrong?

  13. Question 13 of 20 · Multiple Choice

    Solve 2x² + 3x - 2 = 0.

  14. Question 14 of 20 · Multiple Choice

    The quadratic formula gives x = (6 ± √(-20))/2. Which is the same solution written as a ± bi?

  15. Question 15 of 20 · Short Answer

    Complete the square to solve x² + 8x - 5 = 0. Show the equation in the form (x - p)² = q.

  16. Question 16 of 20 · Short Answer

    Solve 3x² - 7 = 68 by taking square roots.

  17. Question 17 of 20 · Short Answer

    Solve x² + 6x + 10 = 0. Write the solutions in a ± bi form.

  18. Question 18 of 20 · Short Answer

    Explain why x² + 9 = 0 has no real solutions, and find its complex solutions.

  19. Question 19 of 20 · Short Answer

    Rewrite 3x² + 18x + 12 = 0 in the form (x - p)² = q, state p and q, and solve.

  20. Question 20 of 20 · Short Answer

    A ball's height in feet after t seconds is h = -16t² + 48t. Write and solve an equation to find when the ball is 32 feet high. Do both solutions make sense?

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

Which method should students use to solve a quadratic equation?

Standard HSA.REI.B.4b asks students to choose the method "as appropriate to the initial form of the equation." A quick guide:

  • Inspection when the answer is visible, such as x² = 49 or x² = 0
  • Square roots when a squared expression is isolated, such as 2(x - 3)² = 50
  • Factoring when the trinomial factors easily over the integers
  • Completing the square when a = 1 and b is even, or when you need the (x - p)² = q form
  • The quadratic formula for everything else; it always works
Why do students need to learn completing the square if the quadratic formula always works?

Completing the square is where the quadratic formula comes from, and standard a asks students to derive the formula from the form (x - p)² = q. The same technique is used later to find the vertex of a parabola (HSF.IF.C.8) and the center and radius of a circle from its equation (HSG.GPE.A.1). Students who only memorize the formula miss that connection.

What does "transform into an equation of the form (x - p)² = q that has the same solutions" mean?

Each step in completing the square (dividing by a, moving the constant, adding the same number to both sides) produces an equivalent equation. So the final equation (x - p)² = q has exactly the same solutions as the original. For example, x² - 6x + 2 = 0 and (x - 3)² = 7 are two forms of the same equation, both with solutions 3 ± √7. Once the equation is in this form, you solve it by taking square roots: x = p ± √q.

What is a common mistake when taking square roots?

Forgetting the negative root. If x² = 49, then x = 7 or x = -7. Remind students that √49 means only 7, so solving requires writing ±√49. A second common error is taking the square root of each term separately, for example turning x² + 9 = 25 into x + 3 = 5. Students should isolate the squared expression first: x² = 16, so x = ±4.

How do I write complex solutions in the form a ± bi?

When b² - 4ac is negative, rewrite the square root using i: √(-36) = 6i and √(-20) = 2i√5. Then divide both parts of the numerator by 2a. For 2x² - 2x + 5 = 0, the formula gives (2 ± 6i)/4, which simplifies to 1/2 ± (3/2)i. The real part is a = 1/2 and the imaginary part is b = 3/2. Students should not leave a negative number under the radical sign.

Is HSA.REI.B.4 taught in Algebra 1 or Algebra 2?

Many course sequences teach the real-number methods (inspection, square roots, factoring, completing the square, and the formula) in Algebra I, and the complex solutions a ± bi in Algebra II, together with the complex number standards HSN.CN.A.1 and HSN.CN.C.7. HSA.REI.B.4 covers both. In Algebra I, students often stop at "no real solutions" when the discriminant is negative; the full standard asks them to write the complex solutions as well.

What is the discriminant, and how does it help?

The discriminant is b² - 4ac, the expression under the square root in the quadratic formula. If it is positive, there are two different real solutions. If it is zero, there is one repeated real solution. If it is negative, there are two complex solutions of the form a ± bi, and the graph of y = ax² + bx + c does not cross the x-axis. Computing it first tells students what kind of answer to expect.

Why do we divide by a before completing the square?

The pattern x² + bx + (b/2)² = (x + b/2)² only works when the coefficient of x² is 1. For 2x² + 12x - 5 = 0, divide every term by 2 first: x² + 6x - 5/2 = 0. Students who skip this step often write 2x² + 12x + 36, which is not a perfect square. Because a ≠ 0 in any quadratic equation, dividing by a is always allowed.

Is HSA.REI.B.4 on the SAT?

Yes. Solving quadratic equations, including by factoring and the quadratic formula, is part of the Advanced Math domain of the digital SAT. Questions may ask for the solutions, the sum of the solutions, or the number of real solutions. Complex solutions written as a ± bi are less typical there than in an Algebra II course.

How should students check their solutions?

Substitute each solution into the original equation, not into a later step, because an error in an earlier step would carry through. For irrational or complex solutions, a calculator check of the decimal value works for real numbers, and for complex numbers students can expand and use i² = -1. For example, for x = 1 + 3i in x² - 2x + 10 = 0: (1 + 3i)² = -8 + 6i, and -8 + 6i - 2 - 6i + 10 = 0.