Quadratic Equations: Methods of Solution and Properties

Quadratic Equations

Definition and Standard Form

A quadratic equation is an algebraic equation of the second degree, meaning the highest exponent of the variable is 2. The standard form of a quadratic equation is:

• Standard Form: $a x^2 + b x + c = 0$

• Where a, b, and c are real numbers, and a \neq 0.

Examples:

• $x^2 - 4x + 5 = 0$ (a = 1, b = -4, c = 5)

• $-3x^2 + 7x - 8 = 0$ (a = -3, b = 7, c = -8)

Types of Quadratic Expressions

• Monomial: $z^2$

• Binomial: $y^2 - 9$

• Trinomial: $2x^2 + 7x - 5$

Factoring Quadratic Equations

Factoring a Difference of Squares

The difference of squares formula is:

• $A^2 - B^2 = (A - B)(A + B)$

Always factor out the greatest common factor (GCF) first.

Example: Factor $9x^2 - 49$

• $9x^2 - 49 = (3x)^2 - 7^2 = (3x - 7)(3x + 7)$

Zero-Product Principle

If the product of two expressions is zero, then at least one of the factors must be zero:

• If $AB = 0$, then $A = 0$ or $B = 0$

Example: If $(x - 1)(x + 2) = 0$, then $x - 1 = 0$ or $x + 2 = 0$, so $x = 1$ or $x = -2$.

Methods to Solve Quadratic Equations

Overview of Solution Methods

There are four main methods to solve quadratic equations:

• Factoring

• Square Root Property

• Completing the Square

• Using the Quadratic Formula

Factoring (ac Method)

To factor a quadratic equation $a x^2 + b x + c$:

1. Factor out the GCF.

2. Find two numbers whose product is $a \cdot c$ and whose sum is $b$.

3. Rewrite the middle term using these numbers.

4. Group and factor each part.

5. Apply the zero-product principle to solve for $x$.

Example: Solve $x^2 - 2x - 3 = 0$ by factoring.

• Find numbers: $(-3) \cdot 1 = -3$, $-3 + 1 = -2$

• Rewrite: $x^2 - 3x + x - 3 = 0$

• Group: $(x^2 - 3x) + (x - 3) = 0$

• Factor: $x(x - 3) + 1(x - 3) = 0$

• Combine: $(x - 3)(x + 1) = 0$

• Solutions: $x = 3$, $x = -1$

Square Root Property

For equations of the form $u^2 = d$:

• $u = \pm \sqrt{d}$

Steps:

1. Isolate the squared term.

2. Take the square root (include $\pm$).

3. Isolate the variable.

4. Check solutions.

Example: Solve $9x^2 = 16$

• $x^2 = \frac{16}{9}$

• $x = \pm \frac{4}{3}$

Completing the Square

Some quadratic equations cannot be factored easily. Completing the square rewrites the equation as a perfect square trinomial.

1. Move the constant to the right side.

2. Divide both sides by the coefficient of $x^2$.

3. Add $\left(\frac{b}{2a}\right)^2$ to both sides.

4. Factor the left side as a square.

5. Take the square root of both sides.

6. Solve for $x$.

Example: Solve $2x^2 - 8x - 24 = 0$ by completing the square.

• Move constant: $2x^2 - 8x = 24$

• Divide by 2: $x^2 - 4x = 12$

• Add $(-4/2)^2 = 4$ to both sides: $x^2 - 4x + 4 = 16$

• Factor: $(x - 2)^2 = 16$

• Take square root: $x - 2 = \pm 4$

• Solutions: $x = 6$, $x = -2$

Quadratic Formula

The quadratic formula can solve any quadratic equation:

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

Example: Solve $2x^2 + 9x - 5 = 0$

• $a = 2$, $b = 9$, $c = -5$

• $x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}$

• $x = \frac{-9 \pm \sqrt{121}}{4}$

• $x = \frac{-9 \pm 11}{4}$

• Solutions: $x = \frac{1}{2}$, $x = -5$

Discriminant and Types of Solutions

Discriminant

The discriminant of a quadratic equation is $D = b^2 - 4ac$.

• If $D > 0$: Two distinct real solutions (graph crosses x-axis twice).

• If $D = 0$: One real solution (graph touches x-axis once).

• If $D < 0$: Two distinct imaginary solutions (graph does not touch x-axis).

Summary of Methods

Method	Can be Used	Comments
Factoring	Sometimes	Use if constant term is zero or factors are easily determined
Square Root Property	Sometimes	Use for equations where b = 0
Quadratic Formula or Completing the Square	Always	Always gives accurate solutions; other methods may be easier in some cases