1Cademy - Solving $$4x^2 - 20x = -25$$ Using the Quadratic Formula

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How to Solve a Quadratic Equation Using the Quadratic Formula

Example

Solving $4x^2 - 20x = -25$ Using the Quadratic Formula

Solve $4x^2 - 20x = -25$ by applying the Quadratic Formula. This example demonstrates the outcome when the expression under the square root equals zero, producing exactly one solution (a double root).

Get standard form. Add $25$ to both sides:

$4x^2 - 20x + 25 = 0$

Step 1 — Identify $a$ , $b$ , $c$ . Here $a = 4$ , $b = -20$ , and $c = 25$ .

Step 2 — Substitute into the Quadratic Formula:

$x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(4)(25)}}{2(4)}$

Step 3 — Simplify. The double negative gives $-(-20) = 20$ . Inside the square root: $(-20)^2 = 400$ and $4(4)(25) = 400$ , so $400 - 400 = 0$ . Since $\sqrt{0} = 0$ :

$x = \frac{20 \pm 0}{8} = \frac{20}{8} = \frac{5}{2}$

There is only one solution: $x = \frac{5}{2}$ .

The trinomial $4x^2 - 20x + 25$ is a perfect square trinomial — it factors as $(2x - 5)^2$ . When a quadratic equation has a perfect square trinomial equal to zero, the expression $b^2 - 4ac$ under the square root evaluates to zero. Because $\pm 0 = 0$ , the "plus or minus" produces only one value rather than two distinct solutions. This connects to the Zero Product Property: the equation $(x - 3)^2 = 0$ has only one solution $x = 3$ , and the Quadratic Formula confirms this same outcome algebraically.

Updated 2026-04-21

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OpenStax Elementary Algebra

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