Quadratic Equations: Methods, Properties, and Applications

Quadratic Equations

Introduction

Quadratic equations are a fundamental topic in algebra and precalculus, appearing in many mathematical models and real-world applications. This section covers the definition, solution methods, and properties of quadratic equations, as well as their applications.

Definition of a Quadratic Equation

• Quadratic Equation: An equation that can be written in the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

• Also called a second-degree polynomial equation in $x$.

The Zero-Product Principle

• To solve a quadratic equation by factoring, use the zero-product principle:

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

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

Solving Quadratic Equations by Factoring

Factoring is an efficient method when the quadratic can be easily factored.

1. Rewrite the equation in the form $ax^2 + bx + c = 0$.

2. Factor the quadratic expression completely.

3. Set each factor containing a variable equal to zero (apply the zero-product principle).

4. Solve the resulting equations.

5. Check the solutions in the original equation.

Example:

• Solve $2x^2 + x = 1$ by factoring.

• Step 1: $2x^2 + x - 1 = 0$

• Step 2: $(2x - 1)(x + 1) = 0$

• Step 3: $2x - 1 = 0$ or $x + 1 = 0$

• Step 4: $x = \frac{1}{2}$ or $x = -1$

• Step 5: Check both solutions in the original equation.

Solving Quadratic Equations by the Square Root Property

This method is useful for equations of the form $u^2 = d$.

• Square Root Property: If $u$ is an algebraic expression and $d$ is a nonzero real number, then $u^2 = d$ has exactly two solutions:

• $u = \sqrt{d}$ or $u = -\sqrt{d}$, equivalently $u = \pm \sqrt{d}$.

Example:

• Solve $5x^2 + 45 = 0$

• $5x^2 = -45$

• $x^2 = -9$

• $x = \pm \sqrt{-9} = \pm 3i$

Completing the Square

Completing the square transforms a quadratic equation into a perfect square trinomial, making it easier to solve.

• For $x^2 + bx$, add $\left(\frac{b}{2}\right)^2$ to both sides:

• $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Example:

• Solve $x^2 + 4x - 1 = 0$ by completing the square.

• $x^2 + 4x = 1$

• $x^2 + 4x + 4 = 1 + 4$

• $(x + 2)^2 = 5$

• $x + 2 = \pm \sqrt{5}$

• $x = -2 \pm \sqrt{5}$

The Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation.

• For $ax^2 + bx + c = 0$ ($a \neq 0$), the solutions are:

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

Example:

• Solve $2x^2 + 2x - 1 = 0$

• $a = 2$, $b = 2$, $c = -1$

• $x = \frac{-2 \pm \sqrt{(2)^2 - 4(2)(-1)}}{2(2)}$

• $x = \frac{-2 \pm \sqrt{4 + 8}}{4}$

• $x = \frac{-2 \pm \sqrt{12}}{4}$

• $x = \frac{-2 \pm 2\sqrt{3}}{4}$

• $x = \frac{-1 \pm \sqrt{3}}{2}$

The Discriminant

The discriminant helps determine the number and type of solutions for a quadratic equation.

• The discriminant is the quantity $b^2 - 4ac$ in the quadratic formula.

• If the discriminant is positive: two unequal real solutions.

• If the discriminant is zero: one real (repeated) solution.

• If the discriminant is negative: two imaginary solutions.

Example:

• For $2x^2 - 7x - 4 = 0$:

• $b^2 - 4ac = (-7)^2 - 4(2)(-4) = 49 + 32 = 81$

• Since 81 is positive, there are two real solutions.

Application of Quadratic Equations

Quadratic equations are used to model real-world phenomena, such as blood pressure as a function of age.

• Given $P = 0.014A^2 + 0.05A + 107$, find $A$ when $P = 115$.

• Set up the equation: $115 = 0.014A^2 + 0.05A + 107$

• Rearrange: $0.014A^2 + 0.05A - 8 = 0$

• Apply the quadratic formula:

$A = \frac{-0.05 \pm \sqrt{(0.05)^2 - 4(0.014)(-8)}}{2(0.014)}$

• Calculate: $A \approx 26$ (positive solution, as age cannot be negative)

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 $u^2 = d$	$x^2 = 9$
Completing the Square	When factoring is difficult or to derive the quadratic formula	$x^2 + 6x + 5 = 0$
Quadratic Formula	Always works for any quadratic equation	$2x^2 + 3x - 2 = 0$