1Cademy - Solving $$x^2 + 8x = 48$$ by Completing the Square

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Completing the Square

Example

Solving $x^2 + 8x = 48$ by Completing the Square

Solve $x^2 + 8x = 48$ by applying the six-step completing-the-square procedure.

Step 1 — Isolate the variable terms. The variable terms $x^2 + 8x$ are already on the left and the constant $48$ is on the right, so no rearranging is needed.

Step 2 — Find $\left(\frac{1}{2} \cdot b\right)^2$ and add it to both sides. The coefficient of $x$ is $8$ , so $b = 8$ . Compute half of $8$ : $\frac{1}{2} \cdot 8 = 4$ . Square the result: $4^2 = 16$ . Add $16$ to both sides:

$x^2 + 8x + 16 = 48 + 16$

Step 3 — Factor the perfect square trinomial. The left side factors as a binomial square:

$(x + 4)^2 = 64$

Step 4 — Apply the Square Root Property:

$x + 4 = \pm\sqrt{64}$

Step 5 — Simplify the radical and solve. Since $64$ is a perfect square ( $8^2 = 64$ ):

$x + 4 = \pm 8$

Write as two equations and solve each:

$x + 4 = 8 \implies x = 4$

$x + 4 = -8 \implies x = -12$

Step 6 — Check both solutions:

For $x = 4$ : $4^2 + 8(4) = 16 + 32 = 48$ ✓

For $x = -12$ : $(-12)^2 + 8(-12) = 144 - 96 = 48$ ✓

The solutions are $x = 4$ and $x = -12$ . This example demonstrates the simplest case of solving by completing the square: the variable terms are already isolated, the coefficient of $x$ is a positive even integer, and the resulting constant on the right side is a perfect square, so the solutions are integers.

Updated 2026-04-21

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OpenStax Elementary Algebra

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