Quadratic Equations: Solving Methods and the Discriminant

Quadratic Equations and Their Solutions

Introduction to Quadratic Equations

A quadratic equation is any equation that can be written in the standard form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a ≠ 0. Quadratic equations are fundamental in algebra and can be solved using several methods.

Methods for Solving Quadratic Equations

There are four main methods for solving quadratic equations:

• Factoring

• Square Root Property

• Completing the Square

• Quadratic Formula

Factoring

• Use when the quadratic can be factored into two binomials.

• Set each factor equal to zero and solve for x.

• Check solutions by substituting back into the original equation.

• Example: $x^2 + 2x - 3 = 0$ factors to $(x + 3)(x - 1) = 0$ so $x = -3$ or $x = 1$.

Square Root Property

• Use when the equation is in the form $x^2 = k$.

• Take the square root of both sides: $x = ext{±} oot{k}$.

• Example: $x^2 = 9$ gives $x = ext{±}3$.

Completing the Square

• Use when the leading coefficient is 1 and factoring is difficult.

• Rewrite the equation so one side is a perfect square trinomial.

• Example: $x^2 + 6x + 9 = 0$ can be written as $(x + 3)^2 = 0$ so $x = -3$.

Quadratic Formula

• Works for any quadratic equation in standard form.

• The formula is:

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

• Plug in values for a, b, and c from the equation.

• Simplify to find the solutions for x.

• Example: For $2x^2 + 3x - 2 = 0$, $a = 2$, $b = 3$, $c = -2$:

$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$ So $x = \frac{2}{4} = \frac{1}{2}$ and $x = \frac{-8}{4} = -2$.

Comparison of Solving Methods

Method	When to Use	Steps
Factoring	Equation factors easily	Factor, set each factor to zero, solve
Square Root Property	No linear term, form $x^2 = k$	Take square root of both sides
Completing the Square	Leading coefficient is 1, not easily factored	Rewrite as perfect square, solve
Quadratic Formula	Any quadratic equation	Plug into formula, simplify

The Discriminant

Understanding the Discriminant

The discriminant is the expression under the square root in the quadratic formula:

$b^2 - 4ac$

The discriminant determines the number and type of solutions to a quadratic equation:

• Positive ($b^2 - 4ac > 0$): Two real solutions

• Zero ($b^2 - 4ac = 0$): One real solution

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

Example: Determining Solutions Using the Discriminant

• For $2x^2 + 3x - 2 = 0$, $b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25$ (Positive: 2 real solutions)

• For $4x^2 + 2x + 2 = 0$, $b^2 - 4ac = 2^2 - 4(4)(2) = 4 - 32 = -28$ (Negative: 2 imaginary solutions)

• For $x^2 - 10x + 25 = 0$, $b^2 - 4ac = (-10)^2 - 4(1)(25) = 100 - 100 = 0$ (Zero: 1 real solution)

Practice Problems

• Solve $3x^2 + 4x + 1 = 0$ using the quadratic formula.

• Solve $2x^2 - 3x = -3$ using the quadratic formula.

• Determine the number and type of solutions for $x^2 + 8x + 16 = 0$ using the discriminant.

• Determine the number and type of solutions for $4x^2 + 5x + 5 = 0$ using the discriminant.

Summary: Quadratic equations can be solved using several methods, with the quadratic formula being universally applicable. The discriminant provides a quick way to determine the number and type of solutions without fully solving the equation.