Quadratic Equations

1. Definition

A quadratic equation is a second-order polynomial equation in a single variable x, with a standard form:

ax² + bx + c = 0

Where:

x is the variable.
a, b, and c are constants, with a ≠ 0.

2. Number of Solutions

When solving quadratic equations, we generally find two solutions because of the squared term. However, depending on the discriminant (the part under the square root in the quadratic formula), we might have one real solution or no real solutions.

3. Methods of Solving Quadratic Equations

Sum/Product Rule: Factoring the quadratic equation into two binomials.
Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Completing the Square: Rewriting the equation to form a perfect square trinomial.

Sum/Product Rule

Practice Questions

1. Solve the equation: \( x^2 - 5x + 6 = 0 \)

2. Solve the equation: \( x^2 - x - 12 = 0 \)

3. Solve the equation: \( 2x^2 - 7x + 3 = 0 \)

4. Solve the equation: \( 3x^2 - 2x - 8 = 0 \)

Quadratic Formula

Practice Questions

Note: Give your answers to 3 significant figures.

1. Solve the equation: \( x^2 - 2x + 1 = 0 \)

2. Solve the equation: \( x^2 - x - 1 = 0 \)

3. Solve the equation: \( 2x^2 - 4x + 3 = 0 \)

Completing the Square

Practice Questions

1. Solve the equation: \( x^2 + 4x + 4 = 0 \)

2. Solve the equation: \( x^2 + 6x + 10 = 0 \)

3. Solve the equation: \( x^2 - 4x + 7 = 0 \)

4. Solve the equation: \( x^2 + 1 = 0 \)