1Cademy - Solving $$3x^2 - 12x - 15 = 0$$ by Completing the Square

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Completing the Square

Example

Solving $3x^2 - 12x - 15 = 0$ by Completing the Square

Solve $3x^2 - 12x - 15 = 0$ by completing the square, demonstrating the procedure when the leading coefficient is not $1$ but can be factored out as a GCF.

Preliminary step — Make the leading coefficient $1$ . The coefficient of $x^2$ is $3$ , which divides evenly into all three terms. Factor out $3$ :

$3(x^2 - 4x - 5) = 0$

Divide both sides by $3$ :

$x^2 - 4x - 5 = 0$

Step 1 — Isolate the variable terms. Add $5$ to both sides to move the constant to the right:

$x^2 - 4x = 5$

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

$x^2 - 4x + 4 = 5 + 4$

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

$(x - 2)^2 = 9$

Step 4 — Apply the Square Root Property:

$x - 2 = \pm\sqrt{9}$

Step 5 — Simplify and solve. Since $9$ is a perfect square ( $3^2 = 9$ ):

$x - 2 = \pm 3$

Write as two equations and solve each:

$x - 2 = 3 \implies x = 5$

$x - 2 = -3 \implies x = -1$

Step 6 — Check both solutions:

For $x = 5$ : $3(5)^2 - 12(5) - 15 = 75 - 60 - 15 = 0$ ✓

For $x = -1$ : $3(-1)^2 - 12(-1) - 15 = 3 + 12 - 15 = 0$ ✓

The solutions are $x = 5$ and $x = -1$ . This example introduces a preliminary step not present in earlier completing-the-square problems: the leading coefficient $3$ must be removed before the standard procedure can begin. Because $3$ divides evenly into all three coefficients ( $3$ , $-12$ , and $-15$ ), factoring it out and dividing both sides by $3$ reduces the equation to the familiar form $x^2 + bx + c = 0$ with a leading coefficient of $1$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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