Quadratic Functions and Equations: Principles, Methods, and Applications

Quadratic Functions and Equations

Introduction

Quadratic functions and equations are foundational topics in precalculus, with wide applications in mathematics and real-world modeling. This section covers the definition, properties, and solution methods for quadratic equations, including factoring, square roots, completing the square, and the quadratic formula.

Quadratic Equations

Definition and Standard Form

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

• Standard Form: The expression $ax^2 + bx + c = 0$ is called the standard form of a quadratic equation.

Quadratic Functions

Definition and Zeros

• Quadratic Function: A function $f$ that can be written as $f(x) = ax^2 + bx + c$, where $a eq 0$.

• Zeros of a Quadratic Function: The values of $x$ for which $f(x) = 0$. These are the solutions to the equation $ax^2 + bx + c = 0$.

• Quadratic functions can have real-number zeros or imaginary-number zeros, depending on the discriminant.

Equation-Solving Principles

Principle of Zero Products

• If $ab = 0$, then $a = 0$ or $b = 0$.

• This principle is used when factoring quadratic equations.

Principle of Square Roots

• If $x^2 = k$, then $x = \\sqrt{k}$ or $x = -\\sqrt{k}$.

• This principle is used when the quadratic equation can be rewritten as a perfect square.

Methods for Solving Quadratic Equations

Factoring

• Express the quadratic equation in standard form.

• Factor the quadratic expression.

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

Example:

Solve $2x^2 - x = 3$.

• Rewrite: $2x^2 - x - 3 = 0$

• Factor: $(x + 1)(2x - 3) = 0$

• Set each factor to zero: $x + 1 = 0$ or $2x - 3 = 0$

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

Using Square Roots

• Isolate the squared term.

• Apply the principle of square roots.

Example:

Solve $2x^2 - 10 = 0$.

• Rewrite: $2x^2 = 10$

• Divide: $x^2 = 5$

• Apply square roots: $x = \\sqrt{5}$ or $x = -\\sqrt{5}$

Completing the Square

• Isolate variable terms on one side and arrange in descending order.

• Divide by the coefficient of $x^2$ if necessary.

• Find half the coefficient of $x$, square it, and add to both sides.

• Express one side as a square of a binomial.

• Use the principle of square roots to solve.

Steps for Completing the Square:

1. Isolate terms with variables.

2. Divide by the coefficient of the squared term if not 1.

3. Complete the square by adding $(\\frac{b}{2a})^2$ to both sides.

4. Express as $(x + d)^2 = k$.

5. Solve for $x$ using square roots.

The Quadratic Formula

General Solution

• The solutions to $ax^2 + bx + c = 0$ are given by:

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

• This formula can be used to solve any quadratic equation.

Example:

Solve $3x^2 + 2x = 7$.

• Rewrite: $3x^2 + 2x - 7 = 0$

• $a = 3$, $b = 2$, $c = -7$

• Apply formula: $x = \\frac{-2 \\pm \\sqrt{2^2 - 4 \\cdot 3 \\cdot (-7)}}{2 \\cdot 3}$

• Exact solutions: $x = \\frac{-2 \\pm \\sqrt{88}}{6}$

• Approximate solutions: $x \\\approx -1.897$ and $x \\\approx 1.230$

The Discriminant

Nature of Solutions

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

• It determines the nature of the solutions:

Discriminant Value	Nature of Solutions
$b^2 - 4ac > 0$	Two distinct real-number solutions
$b^2 - 4ac = 0$	One real-number solution (a repeated root)
$b^2 - 4ac < 0$	Two complex (imaginary) solutions, complex conjugates

Equations Reducible to Quadratic Form

Definition and Method

• Some equations can be transformed into quadratic form by substitution.

• Let $u = x^n$ (where $n$ is an integer), rewrite the equation in terms of $u$, solve for $u$, then back-substitute and solve for $x$.

Example:

Solve $x^4 - 5x^2 + 4 = 0$.

• Let $u = x^2$, so $x^4 = (x^2)^2 = u^2$.

• Rewrite: $u^2 - 5u + 4 = 0$

• Factor: $(u - 1)(u - 4) = 0$

• Solutions: $u = 1$ or $u = 4$

• Back-substitute: $x^2 = 1$ or $x^2 = 4$

• Final solutions: $x = \\pm 1$, $x = \\pm 2$

Applications of Quadratic Equations

Modeling Real-World Problems

• Quadratic equations are used to model various real-world scenarios, such as population growth, projectile motion, and business applications.

• Applied problems can be translated into quadratic equations and solved using the methods above.

Example:

Estimating magazine closures: The function $m(x) = 34x^2 - 59x + 81$ estimates the number of magazine closures $x$ years after 2012. To find when closures reach 99:

• Set $m(x) = 99$

• Equation: $34x^2 - 59x + 81 = 99$

• Rewrite: $34x^2 - 59x - 18 = 0$

• Apply quadratic formula: $x = \\frac{59 \\pm \\sqrt{(-59)^2 - 4 \\cdot 34 \\cdot (-18)}}{2 \\cdot 34}$

• Solutions: $x \\\approx 2$ or $x \\\approx -0.3$

• Since $x$ must be positive, closures reach 99 in 2 years after 2012, i.e., in 2014.

Additional info: The above examples and explanations provide a comprehensive overview of quadratic equations and their applications, suitable for precalculus students preparing for exams.