Algebra • Step-by-Step

Solve 2x² + 5x - 3 = 0 with Steps

Step-by-step solution for 2x² + 5x - 3 = 0 when leading coefficient a ≠ 1. Factors into (2x - 1)(x + 3) = 0 yielding x = 0.5 and x = -3.

Problem Statement

2x^2 + 5x - 3 = 0

Final Result

x_1 = -3, \quad x_2 = \frac{1}{2}

Formula Required: Quadratic Formula with Leading Coefficient

Discriminant: 25 - (-24) = 49. √49 = 7.

x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)} = \frac{-5 \pm \sqrt{49}}{4}

5 Steps

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

2x^2 + 5x - 3 = 0

Apply the $ac$ factoring method: find two numbers that multiply to $ac = (2)(-3) = -6$ and add up to $b = 5$.

(-1) \times (6) = -6, \quad (-1) + (6) = 5

Split the middle term $5x$ into $-1x + (6)x$ and factor by grouping.

2x^2 - 1x + 6x - 3 = 0 \implies (2x - 1)(x + 3) = 0

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

(2x - 1) = 0 \implies x_1 = 0.5 \quad \text{or} \quad (x + 3) = 0 \implies x_2 = -3

Verify each root by substituting back into the original equation.

\text{Check } x = 0.5: \; 2(0.5)^2 + 5(0.5) - 3 = 0.5 + 2.5 - 3 = 0 \;\checkmark \\[6pt] \text{Check } x = -3: \; 2(-3)^2 + 5(-3) - 3 = 18 - 15 - 3 = 0 \;\checkmark

Interactive Calculator

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

x = 1/2 (or 0.5) and x = -3.