1Cademy - Solving $$x^2 - 6x + 5 = 0$$ Using the Quadratic Formula

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Quadratic Equation

Example

Solving $x^2 - 6x + 5 = 0$ Using the Quadratic Formula

Solve $x^2 - 6x + 5 = 0$ by applying the Quadratic Formula. This example demonstrates the procedure when the leading coefficient is $1$ and the linear coefficient is negative.

Step 1 — Identify $a$ , $b$ , $c$ . The equation is already in standard form. Here $a = 1$ , $b = -6$ , and $c = 5$ .

Step 2 — Substitute into the Quadratic Formula:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$

Step 3 — Simplify. The double negative gives $-(-6) = 6$ . Inside the square root: $(-6)^2 = 36$ and $4(1)(5) = 20$ , so $36 - 20 = 16$ . Since $\sqrt{16} = 4$ :

$x = \frac{6 \pm 4}{2}$

Split into two solutions:

$x = \frac{6 + 4}{2} = \frac{10}{2} = 5 \qquad \text{or} \qquad x = \frac{6 - 4}{2} = \frac{2}{2} = 1$

Step 4 — Check both solutions:

For $x = 1$ : $(1)^2 - 6(1) + 5 = 1 - 6 + 5 = 0$ ✓

For $x = 5$ : $(5)^2 - 6(5) + 5 = 25 - 30 + 5 = 0$ ✓

The solutions are $x = 5$ and $x = 1$ . When $b$ is negative, the first term in the numerator of the Quadratic Formula becomes positive because negating a negative number yields a positive result — here, $-(-6) = 6$ . Careful handling of this double negative is essential to avoid sign errors.

Updated 2026-04-21

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OpenStax Elementary Algebra

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