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

Q1. Solve the equation: $-5(x-4) + 2 = 11 - (x-3)$

Background

Topic: Linear Equations

This question tests your ability to solve linear equations by isolating the variable and simplifying both sides.

Key Terms and Formulas:

• Linear equation: An equation involving only the first power of the variable.

• Distributive property: $a(b + c) = ab + ac$

• Combining like terms: Grouping similar terms to simplify.

Step-by-Step Guidance

1. Apply the distributive property to expand $-5(x-4)$ and $-(x-3)$.

2. Simplify both sides by combining like terms.

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

4. Isolate $x$ by dividing or subtracting as needed.

Try solving on your own before revealing the answer!

Final Answer: $x = 2$

After expanding and simplifying, you isolate $x$ and solve for its value.

Q2. Solve the equation: $\frac{1}{x-1} = \frac{5x}{x^2+3x-4} - \frac{3}{x+4}$

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions by finding a common denominator and simplifying.

Key Terms and Formulas:

• Rational expression: A fraction with polynomials in the numerator and denominator.

• Common denominator: The least common multiple of denominators.

• Excluded values: Values that make any denominator zero.

Step-by-Step Guidance

1. Factor $x^2+3x-4$ to find common denominators.

2. Rewrite each term with the common denominator.

3. Combine all terms into a single rational equation.

4. Set the numerator equal to zero and solve for $x$ (excluding values that make denominators zero).

Try solving on your own before revealing the answer!

Final Answer: No solution (because $x=1$ is an excluded value)

After simplifying, the only possible solution is excluded due to division by zero.

Q3. Solve for $t$ in the equation: $T = gr + gvt$

Background

Topic: Solving for a Variable

This question tests your ability to isolate a variable in a formula.

Key Terms and Formulas:

• Isolate: Rearranging the equation to solve for the desired variable.

Step-by-Step Guidance

1. Subtract $gr$ from both sides to isolate terms with $t$.

2. Divide both sides by $gv$ to solve for $t$.

Try solving on your own before revealing the answer!

Final Answer: $t = \frac{T - gr}{gv}$

Rearranging the formula gives $t$ in terms of $T$, $g$, $r$, and $v$.

Q4. Simplify the following and write your answers in the form $a + bi$ when possible:

• a. $(7 + 4i) - (9 + 5i)$

• b. $(7 - i)^2$

• c. $i^{35}$

Background

Topic: Complex Numbers

This question tests your ability to perform arithmetic with complex numbers and powers of $i$.

Key Terms and Formulas:

• Complex number: $a + bi$, where $i$ is the imaginary unit ($i^2 = -1$).

• Arithmetic: Addition, subtraction, and multiplication of complex numbers.

• Powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and repeats every four powers.

Step-by-Step Guidance

1. For part a, subtract real and imaginary parts separately.

2. For part b, expand $(7 - i)^2$ using the binomial formula.

3. For part c, use the cyclical nature of powers of $i$ to simplify $i^{35}$.

Try solving on your own before revealing the answer!

Final Answers:

• a. $-2 - i$

• b. $48 - 14i$

• c. $-i$

Each part uses basic complex number arithmetic and properties of $i$.

Q5. Solve the quadratic equations:

• a. $x^2 + 5x = 24$

• b. $x^2 - 24x + 156 = 0$

• c. $-7x + 3 = -5x^2$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by factoring, completing the square, or using the quadratic formula.

Key Terms and Formulas:

• Quadratic equation: $ax^2 + bx + c = 0$

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

• Factoring: Expressing the equation as a product of binomials.

Step-by-Step Guidance

1. For each equation, rewrite in standard form ($ax^2 + bx + c = 0$).

2. Try factoring if possible; otherwise, use the quadratic formula.

3. Calculate the discriminant ($b^2 - 4ac$) to determine the nature of the solutions.

4. Set up the formula for the final calculation.

Try solving on your own before revealing the answer!

Final Answers:

• a. $x = -8, 3$

• b. $x = 12 + 2i\sqrt{3}, 12 - 2i\sqrt{3}$

• c. $x = \frac{7 + i\sqrt{11}}{10}, \frac{7 - i\sqrt{11}}{10}$

Factoring and the quadratic formula are used for each part.

Q6. Evaluate the discriminant and determine the number and type of solutions to the equation: $x^2 - 3x + 11 = 0$

Background

Topic: Discriminant of Quadratic Equations

This question tests your ability to use the discriminant to determine the nature of the roots of a quadratic equation.

Key Terms and Formulas:

• Discriminant: $D = b^2 - 4ac$

• If $D > 0$: Two real solutions; $D = 0$: One real solution; $D < 0$: Two complex solutions.

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ from the equation.

2. Calculate the discriminant $D = b^2 - 4ac$.

3. Interpret the value of $D$ to determine the number and type of solutions.

Try solving on your own before revealing the answer!

Final Answer: $D = -35$, 2 imaginary solutions

The negative discriminant means the equation has two complex (imaginary) solutions.

Q7. Solve the equation: $x^3 + 2x^2 = 9x + 18$

Background

Topic: Cubic Equations

This question tests your ability to solve cubic equations by factoring and using the zero product property.

Key Terms and Formulas:

• Cubic equation: An equation involving $x^3$.

• Zero product property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Factoring: Expressing the equation as a product of factors.

Step-by-Step Guidance

1. Move all terms to one side to set the equation equal to zero.

2. Factor the cubic equation, possibly by grouping.

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

Try solving on your own before revealing the answer!

Final Answer: $x = -3, -2, 3$

Factoring and solving each factor gives three real solutions.

Q8. Solve the equation: $-\frac{3x}{x-6} - \frac{1}{x+1} = \frac{6}{x^2-5x-6}$

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions by finding a common denominator and simplifying.

Key Terms and Formulas:

• Rational expression: A fraction with polynomials in the numerator and denominator.

• Common denominator: The least common multiple of denominators.

• Excluded values: Values that make any denominator zero.

Step-by-Step Guidance

1. Factor $x^2-5x-6$ to find common denominators.

2. Rewrite each term with the common denominator.

3. Combine all terms into a single rational equation.

4. Set the numerator equal to zero and solve for $x$ (excluding values that make denominators zero).

Try solving on your own before revealing the answer!

Final Answer: $x = -\frac{4}{3}, 0$

After simplifying, you solve for $x$ and check for excluded values.

Q9. Solve the equation: $\sqrt{y+18} + 2 = y$

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots by isolating the radical and squaring both sides.

Key Terms and Formulas:

• Radical equation: An equation with a variable inside a square root.

• Isolate the radical: Move terms to get the square root alone.

• Square both sides: Eliminate the radical by squaring.

Step-by-Step Guidance

1. Isolate $\sqrt{y+18}$ by subtracting 2 from both sides.

2. Square both sides to eliminate the radical.

3. Solve the resulting quadratic equation for $y$.

4. Check for extraneous solutions by plugging back into the original equation.

Try solving on your own before revealing the answer!

Final Answer: $y = 7$

Squaring and solving gives the solution, which checks out in the original equation.

Q10. Solve the equation: $4|2t+7| + 2 = 22$

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values by isolating the absolute value and considering both cases.

Key Terms and Formulas:

• Absolute value: $|x|$ is the distance from zero, always non-negative.

• Isolate the absolute value: Move terms to get $|2t+7|$ alone.

• Consider both cases: $2t+7 = a$ and $2t+7 = -a$.

Step-by-Step Guidance

1. Subtract 2 from both sides to isolate $4|2t+7|$.

2. Divide both sides by 4 to get $|2t+7|$ alone.

3. Set up two equations: $2t+7 = a$ and $2t+7 = -a$.

4. Solve each equation for $t$.

Try solving on your own before revealing the answer!

Final Answer: $t = -6, -1$

Solving both cases gives two solutions for $t$.