How to Solve Quadratic Equations

Quadratic equations are equations of the form \[ ax^2 + bx + c = 0 \], where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It forms a U-shaped graph called a parabola when plotted on a coordinate plane.

Methods to Solve Quadratic Equations

There are three common methods to solve quadratic equations:

1. Factoring
2. Completing the Square
3. Using the Quadratic Formula

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials:
\[ (mx + n)(px + q) = 0 \]

Set each factor equal to zero:
\[ mx + n = 0 \quad \text{or} \quad px + q = 0 \]

Solve for \( x \) to find the solutions.

Example:

Solve \[ x^2 – 5x + 6 = 0 \]

Factor the equation:
\[ (x – 2)(x – 3) = 0 \]

Set each factor to zero:
\[ x – 2 = 0 \quad \text{or} \quad x – 3 = 0 \]

Solutions:
\[ x = 2 \quad \text{or} \quad x = 3 \]

2. Completing the Square

This method rewrites the equation in the form:
\[ (x + p)^2 = q \]

Steps:

1. Move the constant term to the other side: \[ ax^2 + bx = -c \]
2. Divide both sides by \( a \) if \( a \neq 1 \).
3. Add \( \left( \frac{b}{2a} \right)^2 \) to both sides to complete the square.
4. Write the left side as a squared binomial.
5. Solve for \( x \) by taking the square root of both sides.

Example:

Solve \[ x^2 + 6x + 5 = 0 \]

Move the constant term:
\[ x^2 + 6x = -5 \]

Complete the square:
\[ x^2 + 6x + 9 = -5 + 9 \]

Simplify:
\[ (x + 3)^2 = 4 \]

Take the square root:
\[ x + 3 = \pm 2 \]

Solutions:
\[ x = -3 + 2 = -1 \]
\[ x = -3 – 2 = -5 \]

3. Quadratic Formula

The quadratic formula solves any quadratic equation:
\[ x = \frac{ -b \pm \sqrt{b^2 – 4ac} }{ 2a } \]

Example:

Solve \[ 2x^2 – 4x – 6 = 0 \]

Identify coefficients:
\[ a = 2, \quad b = -4, \quad c = -6 \]

Substitute into the formula:
\[ x = \frac{ -(-4) \pm \sqrt{ (-4)^2 – 4 \cdot 2 \cdot (-6) } }{ 2 \cdot 2 } \]

Simplify:
\[ x = \frac{ 4 \pm \sqrt{ 16 + 48 } }{ 4 } \]
\[ x = \frac{ 4 \pm \sqrt{64} }{ 4 } \]
\[ x = \frac{ 4 \pm 8 }{ 4 } \]

Solutions:
\[ x = \frac{4 + 8}{4} = 3 \]
\[ x = \frac{4 – 8}{4} = -1 \]

Summary

Quadratic equations can be solved using factoring, completing the square, or the quadratic formula. Practice using these methods to become proficient at solving quadratics!

Practice Problems

1. Solve \[ x^2 – 16 = 0 \]
2. Solve \[ x^2 + 8x + 16 = 0 \]
3. Solve \[ x^2 – 4x + 1 = 0 \] using the quadratic formula.