EZ Math

Algebra 2 · Quadratic equations

How to Solve a Quadratic by Taking Square Roots

Problem

2x2−32=02x^2 - 32 = 0

Answer

x=4, x=−4x = 4,\ x = -4

When a quadratic has no x term, isolate x squared and take the square root of both sides. A square has two roots, one positive and one negative.

Step-by-step solution

  1. Add 32 to both sides.

    2x2=322x^2 = 32

    Why: The x squared term must be alone before taking a root.

  2. Divide both sides by 2.

    x2=16x^2 = 16

    Why: This leaves a single square equal to a number.

  3. Take the square root of both sides.

    x=±16x = \pm\sqrt{16}

    Why: Both a positive and a negative number square to 16.

  4. Simplify and write both solutions.

    x=4,x=−4x = 4,\quad x = -4

    Why: The plus-or-minus sign gives the two answers.

Common mistakes

  • Keeping only x = 4. The negative root is also a solution.
  • Taking the square root before dividing by 2.
  • Writing x = 16. Taking the root and dividing by 2 are different steps.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. x2=81x^2 = 81

    Show answer

    x=9, x=−9x = 9,\ x = -9

  2. 3x2=753x^2 = 75

    Show answer

    x=5, x=−5x = 5,\ x = -5

  3. x2−18=0x^2 - 18 = 0

    Show answer

    x=32, x=−32x = 3\sqrt{2},\ x = -3\sqrt{2}

  4. (x−2)2=16(x - 2)^2 = 16

    Show answer

    x=6, x=−2x = 6,\ x = -2

  5. x2+9=25x^2 + 9 = 25

    Show answer

    x=4, x=−4x = 4,\ x = -4