College Algebra Exam 3 Review – Step-by-Step Study Guidance

Q1. Find the zeros of $f(x) = 6x^2 - 43x + 7$.

Background

Topic: Quadratic Equations and Zeros of Functions

This question tests your ability to find the zeros (roots) of a quadratic function by solving $f(x) = 0$.

Key Terms and Formulas:

• Zero of a function: A value of $x$ for which $f(x) = 0$.

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

Step-by-Step Guidance

1. Set the function equal to zero: $6x^2 - 43x + 7 = 0$.

2. Identify $a$, $b$, and $c$ from the quadratic equation $ax^2 + bx + c = 0$.

3. Write the quadratic formula and substitute the values of $a$, $b$, and $c$.

4. Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots.

Try solving on your own before revealing the answer!

Q2. Find the zeros of $f(x) = x^6 - 4x^4$.

Background

Topic: Factoring and Solving Polynomial Equations

This question tests your ability to factor higher-degree polynomials and find all real zeros.

Key Terms and Formulas:

• Zero of a function: A value of $x$ for which $f(x) = 0$.

• Factoring: Expressing a polynomial as a product of simpler polynomials.

Step-by-Step Guidance

1. Set $f(x) = 0$: $x^6 - 4x^4 = 0$.

2. Factor out the greatest common factor from the expression.

3. Continue factoring the remaining polynomial, if possible.

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

Try solving on your own before revealing the answer!

Q3. Solve for $x$: $3 - \frac{6}{x + 4} = \frac{x}{x + 4}$

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions (fractions with variables in the denominator).

Key Terms and Formulas:

• Rational equation: An equation containing one or more rational expressions.

• Common denominator: The least common multiple of the denominators in the equation.

Step-by-Step Guidance

1. Identify the common denominator for all terms (here, $x + 4$).

2. Multiply both sides of the equation by the common denominator to eliminate fractions.

3. Simplify the resulting equation.

4. Rearrange the equation to isolate $x$ and solve.

Try solving on your own before revealing the answer!

Q4. Solve for $x$: $12 - 5(x + 3) = 4x - 5$

Background

Topic: Linear Equations

This question tests your ability to solve linear equations with variables on both sides.

Key Terms and Formulas:

• Linear equation: An equation of the form $ax + b = cx + d$.

Step-by-Step Guidance

1. Distribute $-5$ to both terms inside the parentheses: $-5(x + 3)$.

2. Combine like terms on each side of the equation.

3. Move all terms containing $x$ to one side and constants to the other.

4. Solve for $x$ by isolating it.

Try solving on your own before revealing the answer!

Q5. Solve for $x$: $2 - \frac{3}{x + 8} = \frac{x + 2}{x + 8}$

Background

Topic: Rational Equations

This question tests your ability to solve equations with rational expressions.

Key Terms and Formulas:

• Rational equation: An equation involving fractions with variables in the denominator.

• Common denominator: The least common multiple of the denominators.

Step-by-Step Guidance

1. Identify the common denominator ($x + 8$) for all terms.

2. Multiply both sides by the common denominator to clear fractions.

3. Simplify the resulting equation.

4. Rearrange and solve for $x$.

Try solving on your own before revealing the answer!

Q6. Perform the indicated operation and write the result in standard form $(a + bi)$: $(8 - 4i)^2$

Background

Topic: Complex Numbers and Operations

This question tests your ability to expand and simplify expressions involving complex numbers.

Key Terms and Formulas:

• Standard form: $a + bi$, where $a$ and $b$ are real numbers.

• $(a - b)^2 = a^2 - 2ab + b^2$

• $i^2 = -1$

Step-by-Step Guidance

1. Expand $(8 - 4i)^2$ using the formula for the square of a binomial.

2. Calculate each term: $8^2$, $2 \times 8 \times 4i$, and $(4i)^2$.

3. Remember to use $i^2 = -1$ when simplifying $(4i)^2$.

4. Combine real and imaginary parts to write the answer in $a + bi$ form.

Try solving on your own before revealing the answer!

Q7. Perform the indicated operation and write the result in standard form $(a + bi)$: $(-3 + 5i)^2$

Background

Topic: Complex Numbers and Operations

This question tests your ability to expand and simplify expressions involving complex numbers.

Key Terms and Formulas:

• Standard form: $a + bi$

• $(a + b)^2 = a^2 + 2ab + b^2$

• $i^2 = -1$

Step-by-Step Guidance

1. Expand $(-3 + 5i)^2$ using the formula for the square of a binomial.

2. Calculate each term: $(-3)^2$, $2 \times -3 \times 5i$, and $(5i)^2$.

3. Use $i^2 = -1$ to simplify $(5i)^2$.

4. Combine real and imaginary parts to write the answer in $a + bi$ form.

Try solving on your own before revealing the answer!

Q8. Write the quadratic formula.

Background

Topic: Quadratic Equations

This question tests your knowledge of the general formula used to solve quadratic equations.

Key Terms and Formulas:

• Quadratic formula: Used to solve $ax^2 + bx + c = 0$.

Step-by-Step Guidance

1. Recall the formula for solving quadratic equations.

2. Write the formula using $a$, $b$, and $c$.

Try writing the formula before revealing the answer!

Q9. Write an expression for the discriminant.

Background

Topic: Quadratic Equations

This question tests your knowledge of the discriminant, which helps determine the nature of the roots of a quadratic equation.

Key Terms and Formulas:

• Discriminant: The part under the square root in the quadratic formula.

Step-by-Step Guidance

1. Recall the quadratic formula and identify the discriminant.

2. Write the expression for the discriminant in terms of $a$, $b$, and $c$.

Try writing the discriminant before revealing the answer!

Q10. Complete the table as in the notes of section 3: Value of Discriminant, Number and Nature of Solutions.

Background

Topic: Discriminant and Nature of Roots

This question tests your understanding of how the value of the discriminant affects the number and type of solutions to a quadratic equation.

Key Terms and Formulas:

• Discriminant: $D = b^2 - 4ac$

• Nature of roots: Real and distinct, real and equal, or complex conjugate roots.

Step-by-Step Guidance

1. Recall the possible values for the discriminant: $D > 0$, $D = 0$, $D < 0$.

2. For each case, state the number and type of solutions.

Try filling in the table before revealing the answer!