Quadratic Equations: Methods of Solution and the Discriminant

Quadratic Equations

Definition and Standard Form

A quadratic equation is an equation of the form

$ax^2 + bx + c = 0$

where a, b, and c are real numbers and $a \neq 0$.

• Degree: The degree of a quadratic equation is 2, as the highest exponent of the variable is 2.

• Quadratic equations may have real or complex (imaginary) solutions.

Solving Quadratic Equations

There are several methods to solve quadratic equations:

1. Factoring

• Express the quadratic as a product of two binomials and set each factor equal to zero.

• If $ad = bc$, and $A$ and $B$ are algebraic expressions, then $AB = 0$ implies $A = 0$ or $B = 0$.

Example:

Solve $x^2 + 5x + 6 = 0$

Factoring: $(x + 2)(x + 3) = 0$

So, $x = -2$ or $x = -3$

2. Square Root Property

• For equations of the form $x^2 = k$, take the square root of both sides.

• $x^2 = k \implies x = \pm\sqrt{k}$

Example:

Solve $x^2 = 9$

$x = \pm 3$

3. Completing the Square

• Rewrite the equation in the form $(x + d)^2 = k$ by adding a constant to both sides.

• If the leading coefficient is not 1, divide both sides by it first.

• Move the constant to the other side.

• Add the square of half the coefficient of $x$ to both sides to complete the square.

Example:

Solve $x^2 + 6x + 5 = 0$

Move 5: $x^2 + 6x = -5$

Add $(6/2)^2 = 9$ to both sides: $x^2 + 6x + 9 = 4$

So, $(x + 3)^2 = 4$

$x + 3 = \pm 2$

$x = -3 \pm 2 \implies x = -1$ or $x = -5$

4. Quadratic Formula

• For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by:

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

This formula is derived by completing the square on the general quadratic equation.

Example:

Solve $2x^2 + x - 2 = 0$

$a = 2$, $b = 1$, $c = -2$

$x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-2)}}{2 \cdot 2} = \frac{-1 \pm \sqrt{1 + 16}}{4} = \frac{-1 \pm \sqrt{17}}{4}$

The Discriminant

The discriminant is the expression under the square root in the quadratic formula: $b^2 - 4ac$.

• If $b^2 - 4ac > 0$: Two distinct real solutions

• If $b^2 - 4ac = 0$: One real solution (a repeated root)

• If $b^2 - 4ac < 0$: Two complex (imaginary) solutions

Value of $b^2 - 4ac$	Real Solutions	Imaginary Solutions
Positive (>0)	2	0
Zero (=0)	1	0
Negative (<0)	0	2

Example:

For $5x^2 + 4x + 2 = 0$, $a = 5$, $b = 4$, $c = 2$

$b^2 - 4ac = 16 - 40 = -24$

There are 0 real solutions and 2 imaginary solutions.

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic factors easily	$x^2 + 5x + 6 = 0$
Square Root Property	When equation is in the form $x^2 = k$	$x^2 = 9$
Completing the Square	When factoring is difficult or to derive the quadratic formula	$x^2 + 6x + 5 = 0$
Quadratic Formula	Any quadratic equation	$2x^2 + x - 2 = 0$

Additional info: These notes cover the main algebraic techniques for solving quadratic equations, including factoring, the square root property, completing the square, and the quadratic formula. The discriminant is used to classify the nature of the solutions.