Algebra • Step-by-Step

Solve x² - 3x - 10 = 0 with Steps

Step-by-step solution for x² - 3x - 10 = 0. Factors into (x - 5)(x + 2) = 0 yielding roots x = 5 and x = -2.

Problem Statement

x^2 - 3x - 10 = 0

Final Result

x_1 = -2, \quad x_2 = 5

Formula Required: Factoring with Opposite Signs

-5 and +2 multiply to -10 and add to -3.

(x - 5)(x + 2) = 0

6 Steps

Write the equation in standard quadratic form $ax^2 + bx + c = 0$.

x^2 - 3x - 10 = 0

Find two numbers that multiply to $c = -10$ and add up to $b = -3$.

(2) \times (-5) = -10, \quad (2) + (-5) = -3

Write the quadratic expression in factored form.

(x + 2)(x - 5) = 0

Set each factor equal to zero and solve for $x$.

x + 2 = 0 \implies x_1 = -2 \quad \text{or} \quad x - 5 = 0 \implies x_2 = 5

Verify each root by substituting back into the original equation.

\text{Check } x = -2: \; (-2)^2 - 3(-2) - 10 = 4 + 6 - 10 = 0 \;\checkmark \\[6pt] \text{Check } x = 5: \; (5)^2 - 3(5) - 10 = 25 - 15 - 10 = 0 \;\checkmark

Structural Insight (Vieta's Formulas): Notice how the sum and product of the roots directly confirm the original coefficients.

\text{Sum of roots: } x_1 + x_2 = -\frac{b}{a} = 3, \quad \text{Product: } x_1 \cdot x_2 = \frac{c}{a} = -10

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

x = 5 and x = -2.