Solve Quadratic Equations by Factoring

This tutorial explains how to solve quadratic equations by factoring. Detailed solutions are included. You can also use our Factor Quadratic Expressions - Step by Step Calculator.

Questions with Solutions

Solve the following quadratic equation:

\(x^2 - 3x = 0\)

Given: \(x^2 - 3x = 0\)
Factor \(x\) from the left-hand side: \(x(x - 3) = 0\)
Set each factor to zero: \(x = 0\) or \(x - 3 = 0\)
Solutions: \(x = 0\) or \(x = 3\)
Check the solutions by substituting back into the original equation.

Solve the quadratic equation:

\(x^2 - 5x + 6 = 0\)

Factor the quadratic: \(x^2 - 5x + 6 = (x + a)(x + b)\)
Find \(a\) and \(b\) such that \(a + b = -5\) and \(ab = 6\). Solution: \(a = -2, b = -3\)
Factorized form: \(x^2 - 5x + 6 = (x - 2)(x - 3)\)
Set each factor to zero: \(x - 2 = 0\) or \(x - 3 = 0\)
Solutions: \(x = 2\) or \(x = 3\)
Verify the solutions by substituting back.

Solve the quadratic equation:

\(2x^2 + x - 21 = 0\)

Attempt to factor: \(2x^2 + x - 21 = (2x + a)(x + b)\)
Find \(a\) and \(b\) such that \(ab = -21\) and \(a + 2b = 1\). Factorization: \(2x^2 + x - 21 = (2x + 7)(x - 3)\)
Set each factor to zero: \(2x + 7 = 0\) or \(x - 3 = 0\)
Solutions: \(x = -\frac{7}{2}\) or \(x = 3\)
Check the solutions in the original equation.

Solve the equation:

\((x - 1)(x + \frac{1}{2}) = -x + 1\)

Rewrite the equation: \((x - 1)(x + \frac{1}{2}) = -(x - 1)\)
Bring all terms to one side: \((x - 1)(x + \frac{1}{2}) + (x - 1) = 0\)
Factor \(x - 1\): \((x - 1)(x + \frac{3}{2}) = 0\)
Set each factor to zero: \(x - 1 = 0\) or \(x + \frac{3}{2} = 0\)
Solutions: \(x = 1\) or \(x = -\frac{3}{2}\)