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Quadratic Equations

Definition

A quadratic equation is a polynomial equation of degree 2, typically written in the standard form:

ax^2 + bx + c = 0, \quad a \neq 0

• \( a, b, c \) are constants.
• \( x \) is the variable.
• \( a \neq 0 \) to ensure it’s quadratic.

Methods to Solve Quadratic Equations

1. Factoring

Rewrite the equation as:

(px + q)(rx + s) = 0

Then, solve for \( x \) by setting each factor to zero:

px + q = 0 \quad \text{or} \quad rx + s = 0

2. Completing the Square

Rewrite in the form:

(x + h)^2 = k

Solve by taking the square root:

x = -h \pm \sqrt{k}

3. Quadratic Formula

The formula to find the roots of a quadratic equation is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where:

b^2 - 4ac

is called the discriminant.

Nature of Roots

The discriminant \( D = b^2 – 4ac \) determines the nature of the roots:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is one repeated real root.
• If \( D < 0 \), the roots are complex.

Properties of Quadratic Equations

• The graph is a parabola.
• If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• The axis of symmetry is given by:
x = -\frac{b}{2a}
• The vertex of the parabola is:
\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)

Examples

Example 1: Factoring Method

Solve the equation \( x^2 – 5x + 6 = 0 \) by factoring.

x^2 - 5x + 6 = (x - 2)(x - 3) = 0

The solutions are:

x = 2 \quad \text{or} \quad x = 3

Example 2: Quadratic Formula

Solve \( 2x^2 + 3x – 2 = 0 \) using the quadratic formula.

x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} x = \frac{-3 \pm \sqrt{25}}{4} x = \frac{-3 + 5}{4} = \frac{1}{2}, \quad x = \frac{-3 - 5}{4} = -2

The solutions are:

x = \frac{1}{2}, \quad x = -2

Practice Problems

Try solving the following quadratic equations:

1. x^2 - 7x + 10 = 0
2. 3x^2 + x - 4 = 0
3. x^2 + 4x + 5 = 0