Chapter P: Prerequisites – Fundamental Concepts of Algebra (P.7 Equations)

Chapter P: Prerequisites – Fundamental Concepts of Algebra

P.7 Equations

This section reviews essential algebraic techniques for solving equations, including linear, rational, absolute value, quadratic, and radical equations. Mastery of these foundational skills is critical for success in precalculus and higher-level mathematics.

Objectives

• Solve linear equations in one variable, including those with fractions.

• Solve rational equations with variables in denominators.

• Solve a formula for a variable.

• Solve equations involving absolute value.

• Solve quadratic equations by factoring, the square root property, completing the square, and the quadratic formula.

• Use the discriminant to determine the number and type of solutions of quadratic equations.

• Determine the most efficient method for solving a quadratic equation.

• Solve radical equations.

Solving Equations

Definition of a Linear Equation

A linear equation in one variable x is an equation that can be written in the form:

$ax + b = 0$

where a and b are real numbers, and a \neq 0.

Generating Equivalent Equations

Equivalent equations have the same solution set. You can generate equivalent equations by:

• Simplifying expressions (removing grouping symbols, combining like terms).

• Adding or subtracting the same value on both sides.

• Multiplying or dividing both sides by the same nonzero value.

• Interchanging the two sides of the equation.

Solving a Linear Equation

1. Simplify each side by removing grouping symbols and combining like terms.

2. Collect all variable terms on one side and constants on the other.

3. Isolate the variable and solve.

4. Check the solution in the original equation.

Example: Solving a Linear Equation

Solve and check: $4(2x + 1) = 29 + 3(2x - 5)$

• Simplify: $8x + 4 = 6x + 14$

• Collect variables: $2x + 4 = 14$

• Isolate: $2x = 10 \implies x = 5$

• Check: $4(2 \times 5 + 1) = 29 + 3(2 \times 5 - 5) \implies 44 = 44$

Solving Linear Equations Involving Fractions

Clear fractions by multiplying both sides by the least common denominator (LCD).

Example:

Solve and check: $\frac{x-3}{4} = \frac{5}{14} - \frac{x+5}{7}$

• LCD is 28. Multiply both sides by 28 and solve: $7x - 21 = 10 - 4x - 20$

• Combine: $11x - 21 = -10 \implies 11x = 11 \implies x = 1$

• Check by substituting $x = 1$ into the original equation.

Solving Rational Equations

A rational equation contains one or more rational expressions. Clear denominators using the LCD.

Example:

Solve: $\frac{6}{x+3} - \frac{5}{x-2} = \frac{-20}{x^2 + x - 6}$

• Factor denominator: $x^2 + x - 6 = (x+3)(x-2)$

• LCD: $(x+3)(x-2)$

• Multiply both sides by LCD and solve: $6(x-2) - 5(x+3) = -20$

• Solve: $x - 27 = -20 \implies x = 7$

Solving a Formula for a Variable

Isolate the desired variable on one side of the equation.

Example:

Solve for $q$: $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$

• Multiply both sides by $pqf$ and solve: $q = \frac{pf}{p-f}$

Equations Involving Absolute Value

The absolute value of $x$, $|x|$, is the distance from $x$ to zero on the number line. An equation $|u| = c$ (where $c > 0$) is equivalent to $u = c$ or $u = -c$.

Examples:

• Solve $|-2x| = 6$: $-2x = 6 \implies x = -3$ or $-2x = -6 \implies x = 3$

• If $|u| = -c$ (where $c > 0$), there is no solution because absolute value cannot be negative.

Definition of a Quadratic Equation

A quadratic equation in $x$ is an equation that can be written in standard form:

$ax^2 + bx + c = 0$

where $a, b, c$ are real numbers and $a \neq 0$.

The Zero-Product Principle

If $AB = 0$, then $A = 0$ or $B = 0$. This principle is used to solve quadratic equations by factoring.

Solving a Quadratic Equation by Factoring

1. Rewrite in standard form ($ax^2 + bx + c = 0$).

2. Factor completely.

3. Set each factor equal to zero and solve.

4. Check solutions in the original equation.

Example:

Solve $2x^2 - 2x = 0$:

• Factor: $2x(x-1) = 0$

• Set factors to zero: $2x = 0 \implies x = 0$, $x-1 = 0 \implies x = 1$

Solving Quadratic Equations by the Square Root Property

If $u^2 = d$, then $u = \sqrt{d}$ or $u = -\sqrt{d}$.

Example:

Solve $3x^2 - 21 = 0$:

• $3x^2 = 21 \implies x^2 = 7 \implies x = \pm \sqrt{7}$

Completing the Square

To solve $x^2 + bx = c$, add $\left(\frac{b}{2}\right)^2$ to both sides to form a perfect square trinomial.

Example:

Solve $x^2 + 4x = 1$:

• Add $\left(\frac{4}{2}\right)^2 = 4$ to both sides: $x^2 + 4x + 4 = 1 + 4$

• $(x+2)^2 = 5 \implies x+2 = \pm \sqrt{5} \implies x = -2 \pm \sqrt{5}$

The Quadratic Formula

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

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

Example:

For $a = 2, b = 2, c = -1$:

• $x = \frac{-2 \pm \sqrt{(2)^2 - 4(2)(-1)}}{2 \times 2} = \frac{-2 \pm \sqrt{4 + 8}}{4} = \frac{-2 \pm 2\sqrt{3}}{4} = \frac{-1 \pm \sqrt{3}}{2}$

The Discriminant

The discriminant of $ax^2 + bx + c = 0$ is $b^2 - 4ac$.

Discriminant Value	Number and Type of Solutions
$b^2 - 4ac > 0$	Two unequal real solutions
$b^2 - 4ac = 0$	One real (repeated) solution
$b^2 - 4ac < 0$	Two imaginary solutions

Solving Radical Equations

A radical equation contains the variable within a root. To solve:

1. Isolate the radical on one side.

2. Raise both sides to the appropriate power to eliminate the radical.

3. Solve the resulting equation.

4. Check all proposed solutions in the original equation (to avoid extraneous solutions).

Additional info: These foundational algebraic techniques are essential for all subsequent topics in precalculus, including functions, graphing, and advanced equation solving.