Quadratic Equations and Inequalities: College Algebra Study Guide

Quadratic Equations and Inequalities

Introduction to Quadratic Equations

Quadratic equations are fundamental in algebra and appear in many mathematical contexts. A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Quadratic equations are also called polynomials of degree 2.

• Standard Form: All terms are on one side, ordered by descending powers of x: ax2 + bx + c = 0.

• Identifying coefficients: In any quadratic equation, a is the coefficient of x2, b is the coefficient of x, and c is the constant term.

• Example: Write -4x2 + x = 8 in standard form: -4x2 + x - 8 = 0. Here, a = -4, b = 1, c = -8.

Factoring Quadratic Equations

Factoring is a common method for solving quadratic equations, especially when the equation can be written in standard form and the coefficients are manageable.

• Factoring Formulas:

  • $(a + b)(a - b) = a^2 - b^2$

  • $(a + b)^2 = a^2 + 2ab + b^2$

  • $(a - b)^2 = a^2 - 2ab + b^2$

  • $(a + b)(a^2 - ab + b^2) = a^3 + b^3$

  • $(a - b)(a^2 + ab + b^2) = a^3 - b^3$

• Factoring Steps:

  1. Write the equation in standard form.

  2. Factor completely.

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

  4. Check solutions in the original equation.

• Example: Solve x2 + x - 6 = 0 by factoring:

  • Factor: (x + 3)(x - 2) = 0

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

  • Solutions: x = -3 and x = 2

The Square Root Property

When a quadratic equation is in the form (x + h)2 = k, the square root property can be used to solve for x. This method is especially useful when factoring is not possible.

• Steps:

  1. Isolate the squared expression.

  2. Take the positive and negative square roots of both sides.

  3. Solve for x.

  4. (Optional) Check solutions.

• Example: Solve x2 - 5 = 0:

  • Isolate: x2 = 5

  • Take square roots: x = ±√5

• Imaginary Roots: If the squared expression equals a negative number, the solutions are complex (imaginary). For example, x2 + 25 = 0 gives x = ±5i.

Completing the Square

Completing the square is a method used to rewrite a quadratic equation in the form (x + h)2 = k, making it easier to solve using the square root property. This method is useful when the equation cannot be easily factored.

• Steps:

  1. Simplify the equation to x2 + bx = c.

  2. Add $\left(\frac{b}{2}\right)^2$ to both sides.

  3. Rewrite the left side as a perfect square trinomial.

  4. Solve using the square root property.

• Example: Solve x2 + 6x = -7 by completing the square:

  • Add $\left(\frac{6}{2}\right)^2 = 9$ to both sides: x2 + 6x + 9 = 2

  • Rewrite: (x + 3)2 = 2

  • Take square roots: x + 3 = ±√2

  • Solutions: x = -3 ± √2

The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation in standard form. It is especially useful when factoring or completing the square is difficult.

• Quadratic Formula:

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

• Steps:

  1. Write the equation in standard form.

  2. Identify a, b, and c.

  3. Plug values into the formula.

  4. Compute and simplify solutions.

• Example: Solve 3x2 + 4x + 1 = 0:

  • a = 3, b = 4, c = 1

  • $x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

  • $x = \frac{-4 \pm \sqrt{16 - 12}}{6}$

  • $x = \frac{-4 \pm 2}{6}$

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

The Discriminant

The discriminant is the expression under the square root in the quadratic formula: b2 - 4ac. It determines the number and type of solutions for a quadratic equation.

• Discriminant: $b^2 - 4ac$

• Interpretation:

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

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

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

• Example: For x2 + 8x + 16 = 0, $b^2 - 4ac = 64 - 64 = 0$ (one real solution).

Choosing a Method to Solve Quadratic Equations

There are several methods to solve quadratic equations, and the best method depends on the form and coefficients of the equation. Sometimes multiple methods are possible.

• Factoring: Use when the equation has obvious factors or c = 0.

• Square Root Property: Use when the equation is in the form (x + h)2 = k or b = 0.

• Completing the Square: Use when the leading coefficient is 1 and b is even.

• Quadratic Formula: Use when factoring is difficult or uncertain.

• Example: For x2 + 3x = 0, factoring is best. For x2 + 6x + 1 = 0, use the quadratic formula.

Summary Table: Methods for Solving Quadratic Equations

Method	Best Used When	Steps
Factoring	Obvious factors, c = 0	Standard form → Factor → Set factors = 0 → Solve
Square Root Property	(x + h)2 = k, b = 0	Isolate squared term → Take ±√ → Solve
Completing the Square	Leading coefficient 1, b even	Simplify → Add (b/2)2 → Factor → Solve
Quadratic Formula	Factoring difficult, unsure method	Standard form → Plug a, b, c → Compute

Additional info: The notes include practice problems and step-by-step examples for each method, reinforcing the concepts and providing opportunities for mastery.