Solving Quadratic Equations by Reasoning

10 min

Narrative

In this Warm-up, students begin to think more abstractly about the process of solving quadratic equations. They recognize that some quadratic equations have one solution and others have two.

All of the equations can be solved by reasoning and do not require formal knowledge of algebraic methods, such as rewriting into factored form or completing the square. For example, for x2−1=3x^2 -1 = 3, students can reason that x2x^2 must be 4 because that is the only number that gives 3 when subtracted by 1.

Finding the solutions of these equations, especially the last few equations, requires perseverance in making sense of problems and of representations (MP1).

Launch

Ask students to evaluate 4⋅44 \boldcdot 4 and -4⋅(-4)\text-4 \boldcdot(\text-4). Make sure they recall that both products are positive 16.

Student Task

How many solutions does each equation have? What are the solution(s)? Be prepared to explain how you know.

  1. x2=9x^2 = 9
  2. x2=0x^2 =0
  3. x2−1=3x^2 -1 = 3
  4. 2x2=502x^2 = 50
  5. (x+1)(x+1)=0(x+1)(x+1)=0
  6. x(x−6)=0x(x-6)=0
  7. (x−1)(x−1)=4(x-1)(x-1)=4

Sample Response

  1. Two solutions: x=3x=3 and x=-3x= \text-3
  2. One solution: x=0x=0
  3. Two solutions: x=2x=2 and x=-2x=\text-2
  4. Two solutions: x=5x=5 and x=-5x=\text-5
  5. One solution: x=-1x=\text-1
  6. Two solutions: x=0x=0 and x=6x=6
  7. Two solutions: x=3x=3 and x=-1x=\text-1
Activity Synthesis (Teacher Notes)

Ask students to share their responses and reasoning for the last four questions. After each student explains, ask the class if they agree or disagree and discuss any disagreements.

Make sure students see that in cases such as x(x−6)=0x(x-6)=0 and (x−1)(x−1)=4(x-1)(x-1)=4, the solutions to each equation may not necessarily be opposites, as was the case in the preceding equations. For example, in the last question, we want to find a number that produces 4 when it is squared. That number can be 2 or -2. If the number is 2, then xx is 3. If the number is -2, then xx is -1.

If time permits, discuss questions such as:

  • “What do you notice about the quadratic equations that have only 1 solution?” (They seem to be written as a factor squared equal to 0.)
  • "In an equation like x(x−6)=0x(x-6)=0, how can we tell that there are two solutions?" (There are two factors—either of which could make the product 0.)
Anticipated Misconceptions

When solving 2x2=502x^2=50, some students may confuse 2x22x^2 with (2x)2(2x)^2 and conclude that the solutions are 502\frac{\sqrt{50}}{2} and -502\text-\frac{\sqrt{50}}{2}. Clarify that 2x22x^2 means 2 times x2x^2, and that the only thing being squared is the xx. If both the 2 and xx are squared, a pair of parentheses is used to group the 2 and the xx so that we know both are being squared.

Standards
Addressing
  • A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A-REI.4.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.
  • A-REI.4.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.
  • A-REI.4.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.
  • HSA-REI.A.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • HSA-REI.B.4.b·Solve quadratic equations by inspection (e.g., for <span class="math">\(x^2 = 49\)</span>), 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 <span class="math">\(a \pm bi\)</span> for real numbers <span class="math">\(a\)</span> and <span class="math">\(b\)</span>.

25 min