College Algebra Study Guide: Quadratics, Exponents, and Radicals

Quadratics, Exponents, and Radicals

Review of Basic Concepts: Simplifying Radicals and Rationalizing Denominators

This section covers the simplification of radical expressions and the process of rationalizing denominators. These are foundational skills for manipulating algebraic expressions involving roots.

• Radical Expression: An expression that contains a root, such as a square root or cube root.

• Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction.

Example:

Simplify $\sqrt{72}$ and rationalize denominators:

• $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

• $\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Key Steps:

1. Factor the number under the radical into perfect squares.

2. Simplify by taking the square root of perfect squares.

3. Multiply numerator and denominator by a radical to rationalize denominators.

Example:

• $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

• $\frac{\sqrt{125}}{7} = \frac{5\sqrt{5}}{7}$

Equations and Inequalities: Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. There are several methods to solve them, including factoring, completing the square, and using the quadratic formula.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Completing the Square: Rewrite the equation in the form $(x + p)^2 = q$ and solve for $x$.

• Quadratic Formula: For any quadratic equation $ax^2 + bx + c = 0$, the solutions are:

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

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real and distinct, real and equal, or complex).

Example:

• Solve $x^2 + 6x + 5 = 0$ by factoring:

• $(x + 5)(x + 1) = 0 \implies x = -5, -1$

• Solve $x^2 + 24x - 18 = 0$ using the quadratic formula:

• $x = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 1 \cdot (-18)}}{2}$

Graphs and Functions: Properties of Quadratic Functions

Quadratic functions have the general form $f(x) = ax^2 + bx + c$. Their graphs are parabolas, and key features include the vertex, axis of symmetry, and intercepts.

• Vertex: The highest or lowest point of the parabola, given by $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$.

• Y-intercept: The point where the graph crosses the y-axis ($x = 0$).

Polynomials and Rational Functions: Simplifying Expressions with Exponents

Expressions with exponents can often be simplified using the laws of exponents. When negative or fractional exponents are present, rewrite them with positive exponents and radicals as needed.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Fractional Exponent: $a^{1/n} = \sqrt[n]{a}$

Example:

• Express $(4x^{-2}y^4)^{1/2}$ in simplest form:

• $(4x^{-2}y^4)^{1/2} = 2x^{-1}y^2 = \frac{2y^2}{x}$

Radical Operations: Addition, Subtraction, and Rationalization

When performing operations with radicals, combine like terms and rationalize denominators as needed.

• Addition/Subtraction: Only like radicals (same index and radicand) can be combined.

• Multiplication: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

• Rationalizing: Multiply numerator and denominator by a radical to eliminate radicals in the denominator.

Example:

• $\sqrt{20} + \sqrt{45} = 2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$

• $\frac{\sqrt{3}}{\sqrt{5}} = \frac{\sqrt{15}}{5}$

Solving Radical Equations

Radical equations are equations in which the variable appears under a radical. To solve, isolate the radical and raise both sides to the appropriate power.

• Isolate the radical on one side of the equation.

• Raise both sides to the power that eliminates the radical.

• Solve the resulting equation and check for extraneous solutions.

Example:

• Solve $\sqrt{5x+9} + 1 = x$:

• Isolate: $\sqrt{5x+9} = x - 1$

• Square both sides: $5x + 9 = (x - 1)^2$

• Solve the quadratic equation for $x$.

Summary Table: Laws of Exponents

Law	Formula	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^2 = 2^5 = 32$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^3$
Power Rule	$(a^m)^n = a^{mn}$	$(y^2)^3 = y^6$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = 1/x^3$
Fractional Exponent	$a^{1/n} = \sqrt[n]{a}$	$8^{1/3} = 2$

Additional info:

• Check for extraneous solutions when solving radical equations by substituting back into the original equation.

• When rationalizing denominators with binomials, multiply by the conjugate if necessary.