Key Points

✓A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0.
✓The solutions to a quadratic equation are called roots or zeros.
✓Quadratic equations can have two, one, or no real solutions.
✓Factoring, the quadratic formula, and completing the square are common methods for solving quadratic equations.
✓The discriminant (b² - 4ac) determines the number of real solutions.

A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants (numbers) and 'a' cannot be zero. Quadratic equations are used to model many real-world situations, such as the path of a projectile or the area of a shape. Solving a quadratic equation means finding the values of 'x' that make the equation true. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method has its own advantages depending on the specific equation.

Worked Example

Solve the quadratic equation x² + 5x + 6 = 0 by factoring.

Step 1: Factor the quadratic expression. Find two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.
Step 2: Rewrite the equation as (x + 2)(x + 3) = 0.
Step 3: Set each factor equal to zero: x + 2 = 0 or x + 3 = 0.
Step 4: Solve for x in each equation: x = -2 or x = -3.

Answer: The solutions are x = -2 and x = -3.

Try It Yourself

1. Solve for x: x² - 4 = 0

2. Solve for x: 2x² + 5x - 3 = 0

3. Solve for x: x² - 8x + 12 = 0

Watch Out For These Mistakes

Forgetting to set the equation to zero before factoring or using the quadratic formula.
Incorrectly applying the quadratic formula (especially the signs).
Making errors when factoring quadratic expressions.