College Algebra Pretest Review: Equations, Simplification, and Polynomial Operations

Equations & Inequalities

Solving Quadratic Equations by Taking Square Roots

Quadratic equations can sometimes be solved directly by isolating the squared term and taking the square root of both sides. This method is efficient when the equation is in the form $ax^2 = k$ or $ax^2 + c = k$.

• Key Steps:

  1. Isolate the squared term on one side of the equation.

  2. Take the square root of both sides, remembering to include both the positive and negative roots.

  3. Solve for the variable.

• Example: Solve $2x^2 - 2 = 30$.

  • Add 2 to both sides: $2x^2 = 32$

  • Divide by 2: $x^2 = 16$

  • Take the square root: $x = \pm 4$

Solving Quadratic Equations by Completing the Square

Completing the square is a method used to solve quadratic equations that are not easily factorable. It involves rewriting the equation in the form $(x + p)^2 = q$.

• Key Steps:

  1. Move the constant term to the other side of the equation.

  2. If the coefficient of $x^2$ is not 1, divide both sides by that coefficient.

  3. Add the square of half the coefficient of $x$ to both sides to complete the square.

  4. Rewrite the left side as a squared binomial.

  5. Take the square root of both sides and solve for $x$.

• Example: Solve $x^2 + 14x + 83 = -7$ by completing the square.

  • Subtract 83: $x^2 + 14x = -90$

  • Add $(\frac{14}{2})^2 = 49$ to both sides: $x^2 + 14x + 49 = -90 + 49$

  • So, $x^2 + 14x + 49 = -41$

  • Rewrite: $(x + 7)^2 = -41$

  • Take square root: $x + 7 = \pm \sqrt{-41}$

  • So, $x = -7 \pm i\sqrt{41}$

Review of Algebra

Simplifying Sums and Differences of Polynomials

Adding and subtracting polynomials involves combining like terms, which are terms with the same variable raised to the same power.

• Key Steps:

  1. Arrange terms in standard form (descending powers of the variable).

  2. Combine coefficients of like terms.

• Example (Sum): $(2m^2 + 7m^3 + 5) + (5m^2 + 8m^3 + 3) + (5m^2 + 3m^3)$

  • Combine $m^3$ terms: $7m^3 + 8m^3 + 3m^3 = 18m^3$

  • Combine $m^2$ terms: $2m^2 + 5m^2 + 5m^2 = 12m^2$

  • Combine constants: $5 + 3 = 8$

  • Final answer: $18m^3 + 12m^2 + 8$

• Example (Difference): $(4m^2 - 7 - 4m^7) - (3 + m^2 - 4m^7)$

  • Distribute the negative: $4m^2 - 7 - 4m^7 - 3 - m^2 + 4m^7$

  • Combine like terms: $(4m^2 - m^2) + (-7 - 3) + (-4m^7 + 4m^7)$

  • Result: $3m^2 - 10$

Multiplying Polynomials

To multiply polynomials, use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second.

• Key Steps:

  1. Multiply each term in the first polynomial by each term in the second.

  2. Combine like terms.

• Example: $(x + 7)(x + 3)$

  • First: $x \cdot x = x^2$

  • Outer: $x \cdot 3 = 3x$

  • Inner: $7 \cdot x = 7x$

  • Last: $7 \cdot 3 = 21$

  • Combine: $x^2 + 10x + 21$

Formulas and Key Concepts

• Quadratic Formula: For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a + b)^2$

Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic is easily factorable	$x^2 - 5x + 6 = 0$
Square Roots	When the equation is in the form $ax^2 = k$	$2x^2 = 18$
Completing the Square	When the quadratic is not easily factorable	$x^2 + 6x + 5 = 0$
Quadratic Formula	Works for any quadratic equation	$x^2 + 4x + 1 = 0$

Additional info: The problems and answer key provided are typical of a College Algebra pretest, covering foundational algebraic skills such as solving quadratic equations, simplifying and combining polynomials, and multiplying polynomials. Mastery of these topics is essential for success in more advanced algebraic concepts.