1Cademy - Solving $$4x^2 = 16x + 84$$ by Factoring

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Quadratic Equation

Example

Solving $4x^2 = 16x + 84$ by Factoring

Solve $4x^2 = 16x + 84$ by writing in standard form, factoring out the numerical GCF, and recognizing that a constant factor cannot equal zero.

Step 1 — Write in standard form. Subtract $16x$ and $84$ from both sides:

$4x^2 - 16x - 84 = 0$

Step 2 — Factor the greatest common factor first. All three terms share a numerical GCF of $4$ :

$4(x^2 - 4x - 21) = 0$

Step 3 — Factor the trinomial. Find two numbers whose product is $-21$ and whose sum is $-4$ . The pair $3$ and $-7$ works: $3 \cdot (-7) = -21$ and $3 + (-7) = -4$ . Therefore:

$4(x - 7)(x + 3) = 0$

Step 4 — Apply the Zero Product Property. The factored equation has three factors: $4$ , $(x - 7)$ , and $(x + 3)$ . However, the first factor is the constant $4$ , and a nonzero constant can never equal zero ( $4 \neq 0$ ). Only the variable factors need to be set equal to zero:

$x - 7 = 0 \quad \text{or} \quad x + 3 = 0$

Step 5 — Solve each equation:

$x = 7 \quad \text{or} \quad x = -3$

The solutions are $x = 7$ and $x = -3$ . When factoring produces a nonzero constant as one of the factors, that constant can be disregarded in the Zero Product Property step because it can never be zero. Only the factors containing the variable contribute solutions to the equation.

Updated 2026-04-21

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OpenStax Elementary Algebra

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