Quadratic Equations and Applications: College Algebra Study Notes

Quadratic Equations and Applications

Introduction to Quadratic Equations

Quadratic equations are fundamental in algebra and appear frequently in mathematical modeling and problem solving. A quadratic equation in the variable x can be written in the general form:

General Form:

$ax^2 + bx + c = 0$

• a, b, c are real numbers, with $a \neq 0$.

• This is a second-degree polynomial equation in x.

• Quadratic equations can be solved using four main methods: factoring, extracting square roots, completing the square, and the quadratic formula.

Methods for Solving Quadratic Equations

There are four principal methods for solving quadratic equations:

1. Factoring

2. Extracting Square Roots

3. Completing the Square

4. Quadratic Formula

Solving Quadratic Equations by Factoring

Zero-Factor Property

The factoring method is based on the Zero-Factor Property:

$ab = 0 \implies a = 0 \text{ or } b = 0$

• Rewrite the quadratic equation so one side equals zero.

• Factor the quadratic expression into a product of two linear factors.

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

Example: Solving by Factoring

• Given: $2x^2 + 7x = 3$

• Write in general form: $2x^2 + 7x - 3 = 0$

• Factor: $(2x - 1)(x + 3) = 0$

• Set each factor to zero: $2x - 1 = 0 \implies x = \frac{1}{2}$; $x + 3 = 0 \implies x = -3$

• Solutions: $x = \frac{1}{2}$ and $x = -3$

Important Notes

• The Zero-Factor Property applies only when the equation is set to zero.

• All terms must be collected on one side before factoring.

• For equations like $(x - 5)(x + 2) = 8$, first expand and move all terms to one side: $x^2 - 3x - 10 = 8 \implies x^2 - 3x - 18 = 0$

Solving Quadratic Equations by Extracting Square Roots

Extracting Square Roots Method

This method is used for equations of the form $u^2 = d$, where $d > 0$.

• Take the square root of both sides: $u = \pm \sqrt{d}$

• This yields two solutions, differing only in sign.

Example: Extracting Square Roots

• Given: $4(x - 3)^2 = 7$

• Divide both sides by 4: $(x - 3)^2 = \frac{7}{4}$

• Extract square roots: $x - 3 = \pm \sqrt{\frac{7}{4}} = \pm \frac{\sqrt{7}}{2}$

• Add 3 to both sides: $x = 3 \pm \frac{\sqrt{7}}{2}$

Solving Quadratic Equations by Completing the Square

Completing the Square Method

Completing the square is useful when the quadratic expression is not easily factorable. The process involves rewriting the equation so that one side is a perfect square trinomial.

• Given: $x^2 + bx + c = 0$

• Move the constant to the other side: $x^2 + bx = -c$

• Add $(\frac{b}{2})^2$ to both sides to complete the square.

• Rewrite as: $(x + \frac{b}{2})^2 = -c + (\frac{b}{2})^2$

• Extract square roots and solve for x.

Example: Completing the Square

• Given: $x^2 + 2x - 6 = 0$

• Add 6 to both sides: $x^2 + 2x = 6$

• Add $(\frac{2}{2})^2 = 1$ to both sides: $x^2 + 2x + 1 = 7$

• Rewrite: $(x + 1)^2 = 7$

• Extract square roots: $x + 1 = \pm \sqrt{7}$

• Subtract 1: $x = -1 \pm \sqrt{7}$

Solving Quadratic Equations Using the Quadratic Formula

The Quadratic Formula

The quadratic formula provides a universal method for solving any quadratic equation:

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

• Applicable to all quadratic equations in the form $ax^2 + bx + c = 0$.

• The expression under the square root, $b^2 - 4ac$, is called the discriminant.

Discriminant and Nature of Solutions

Discriminant ($b^2 - 4ac$)	Number and Type of Solutions
$> 0$	Two distinct real solutions
$= 0$	One repeated real solution
$< 0$	No real solutions (two complex solutions)

Example: Using the Quadratic Formula

• Given: $x^2 + 3x = 9$

• Write in general form: $x^2 + 3x - 9 = 0$

• Identify $a = 1$, $b = 3$, $c = -9$

• Discriminant: $b^2 - 4ac = 9 - 4(1)(-9) = 9 + 36 = 45$

• Apply formula: $x = \frac{-3 \pm \sqrt{45}}{2}$

• Simplify: $x = \frac{-3 \pm 3\sqrt{5}}{2}$

Applications of Quadratic Equations

Modeling Real-Life Problems

Quadratic equations are often used to model problems involving area, projectile motion, and other physical phenomena.

Example: Area of a Square Room

• Let $x$ be the length of each side of a square room with area 144 square feet.

• Equation: $x^2 = 144$

• Solutions: $x = 12$ or $x = -12$

• Only the positive solution is meaningful in this context: $x = 12$ feet.

Example: Dimensions of a Rectangular Room

• Width = $w$ feet, Length = $w + 3$ feet, Area = 154 square feet.

• Equation: $w(w + 3) = 154 \implies w^2 + 3w - 154 = 0$

• Factor: $(w - 11)(w + 14) = 0$

• Solutions: $w = 11$ or $w = -14$

• Width is 11 feet (positive value), Length is 14 feet.

Summary Table: Methods for Solving Quadratic Equations

Method	Form of Equation	Key Steps
Factoring	$ax^2 + bx + c = 0$ (factorable)	Factor, set each factor to zero, solve for $x$
Extracting Square Roots	$u^2 = d$	Take square roots, $u = \pm \sqrt{d}$
Completing the Square	$ax^2 + bx + c = 0$	Rewrite as perfect square, solve for $x$
Quadratic Formula	$ax^2 + bx + c = 0$	Apply $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Additional info: These notes expand on the original slides by providing full definitions, step-by-step examples, and tables for comparison. All equations are presented in LaTeX format for clarity.