9.8.3 Understanding That the Quadratic Formula Is the Combined Steps of Completing the Square

Algebra 19.8.3 Understanding That the Quadratic Formula Is the Combined Steps of Completing the Square

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Here is one solution for solving $x^{2} + 3 x + 8 = 0$ by completing the square, where each step is shown, but numerical expressions are not evaluated.

Original equation $x^{2} + 3 x + 8 = 0$

Step 1 - $x^{2} + 3 x = - 8$

Step 2 - $4 x^{2} + 4 (3 x) = 4 (- 8)$

Step 3 - $(2 x)^{2} + 6 (2 x) = - 32$

Step 4 - $P^{2} + 6 P = - 32$

Step 5 - $P^{2} + 6 P + 3^{2} = - 32 + 3^{2}$

Step 6 - $(P + 3)^{2} = 3^{2} - 32$

Step 7 - $P + 3 = \pm \sqrt{3^{2} - 32}$

Step 8 - $P = - 3 \pm \sqrt{3^{2} - 32}$

Step 9 - $2 x = - 3 \pm \sqrt{3^{2} - 32}$

Step 10 -

$x = \frac{- 3 \pm \sqrt{3^{2} - 32}}{2}$

In Step 2, the equation is multiplied by 4. Why might that be?

Compare your answer:

To make the coefficient of $x^{2}$ a perfect square and the coefficient of $x$ an even number

In Step 5, $3^{2}$ is added to each side. Why might that be?

Compare your answer:

$3^{2}$ completes the square for the expression on the left side. To keep the two sides equal, it needs to be added to the other side as well.

What happened between Step 5 and Step 6?

Compare your answer:

The perfect square is rewritten in factored form, and $- 32 + 3^{2}$ is rewritten as $3^{2} - 32$ .

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- Publication date: Jun 4, 2025
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