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Algebra

Question 1 of 4

The Quadratic Formula

Question

Find the solution of \(x^2 - 4x - 32 = 0\) using the quadratic formula

Show solution Hide solution Fully worked — 8 steps
  1. Check the equation is in the form \(ax^2 + bx + c = 0\)

    Since the equation is in the form \(ax^2 + bx + c = 0\), proceed to step 2.

    \[ x^2 - 4x - 32 = 0 \]
  2. Identify the values of \(a\), \(b\) and \(c\)
    \[ a = 1 \qquad b = -4 \qquad c = -32 \]
  3. Use the formula to find the solutions
    \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
  4. Substitute the values of \(a\), \(b\) and \(c\) and solve
    \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-32)}}{2(1)} \]
  5. Evaluate each part

    \(-(-4) = 4\); \((-4)^2 = 16\) and \(-4(1)(-32) = +128\):

    \[ = \frac{4 \pm \sqrt{16 + 128}}{2} \]
  6. Simplify under the square root

    \(16 + 128 = 144\):

    \[ = \frac{4 \pm \sqrt{144}}{2} \]

    \(\sqrt{144} = 12\):

    \[ = \frac{4 \pm 12}{2} \]
  7. Find the two solutions

    Then

    \[ x = \frac{4 + 12}{2} = 8 \]

    or

    \[ x = \frac{4 - 12}{2} = -4 \]
  8. State the solutions

    So the solutions are \(x = 8\) and \(x = -4\). That is, \(x\) is eight and \(x\) is minus four.