College Algebra Exam 2 Study Guide: Step-by-Step Guidance

Q1. Solve the quadratic equation: $x^2 - 8x - 20 = 0$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations exactly, using factoring, completing the square, or 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 quadratic as a product of two binomials, if possible.

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the equation $x^2 - 8x - 20 = 0$.

2. Check if the quadratic can be factored easily. If not, prepare to use the quadratic formula.

3. Set up the quadratic formula with your values for $a$, $b$, 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!

Final Answer:

$x = 10$ or $x = -2$

Factoring or using the quadratic formula gives these two exact solutions.

Q2. Solve the quadratic equation: $4x^2 - 12 = 0$

Background

Topic: Quadratic Equations (Solving by Square Roots)

This question tests your ability to solve a quadratic equation that can be simplified and solved by isolating $x^2$.

Key Terms and Formulas:

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

• Solving by Square Roots: If $x^2 = k$, then $x = \pm\sqrt{k}$

Step-by-Step Guidance

1. Add 12 to both sides to isolate the $x^2$ term.

2. Divide both sides by 4 to solve for $x^2$.

3. Take the square root of both sides, remembering to consider both the positive and negative roots.

Try solving on your own before revealing the answer!

Final Answer:

$x = \sqrt{3}$ or $x = -\sqrt{3}$

These are the exact solutions after isolating $x^2$ and taking square roots.

Q3. Solve the quadratic equation: $x^2 + 6x = 7$

Background

Topic: Quadratic Equations (Completing the Square or Quadratic Formula)

This question tests your ability to solve a quadratic equation that is not set to zero. You may use completing the square or the quadratic formula.

Key Terms and Formulas:

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

• Completing the Square: Rewriting $x^2 + bx$ as $(x + \frac{b}{2})^2 - (\frac{b}{2})^2$

Step-by-Step Guidance

1. Subtract 7 from both sides to set the equation to zero: $x^2 + 6x - 7 = 0$.

2. Identify $a$, $b$, and $c$ for the quadratic formula, or try factoring if possible.

3. If factoring is not straightforward, set up the quadratic formula with your values.

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

Try solving on your own before revealing the answer!

Final Answer:

$x = 1$ or $x = -7$

These are the exact solutions after factoring or using the quadratic formula.

Q4. Solve the quadratic equation: $-x^2 + 8x + 25 = 0$

Background

Topic: Quadratic Equations (Negative Leading Coefficient)

This question tests your ability to solve a quadratic equation where the leading coefficient is negative.

Key Terms and Formulas:

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

• Factoring: If possible, factor the equation after multiplying both sides by $-1$.

Step-by-Step Guidance

1. Multiply both sides by $-1$ to make the leading coefficient positive: $x^2 - 8x - 25 = 0$.

2. Identify $a$, $b$, and $c$ for the quadratic formula, or try factoring.

3. Set up the quadratic formula with your values.

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

Try solving on your own before revealing the answer!

Final Answer:

$x = 4 + 3\sqrt{3}$ or $x = 4 - 3\sqrt{3}$

These are the exact solutions using the quadratic formula.

Q5. Solve the quadratic equation: $3x^2 + 6 = 10x$

Background

Topic: Quadratic Equations (Rearranging and Solving)

This question tests your ability to rearrange and solve a quadratic equation using factoring or the quadratic formula.

Key Terms and Formulas:

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

• Factoring: If possible, factor the equation after rearranging.

Step-by-Step Guidance

1. Subtract $10x$ from both sides to set the equation to zero: $3x^2 - 10x + 6 = 0$.

2. Identify $a$, $b$, and $c$ for the quadratic formula, or try factoring.

3. Set up the quadratic formula with your values.

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

Try solving on your own before revealing the answer!

Final Answer:

$x = 2$ or $x = 1$

These are the exact solutions after rearranging and solving the quadratic.

Q6. Solve the quadratic equation: $5x^2 - 6x - 27 = 0$

Background

Topic: Quadratic Equations (Quadratic Formula)

This question tests your ability to solve a quadratic equation that may not factor easily, so the quadratic formula is likely needed.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify $a = 5$, $b = -6$, $c = -27$.

2. Set up the quadratic formula with these values.

3. Calculate the discriminant $b^2 - 4ac$.

4. Plug the discriminant and coefficients into the quadratic formula, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Final Answer:

$x = 3$ or $x = -\frac{9}{5}$

These are the exact solutions using the quadratic formula.

Q7. For $f(x) = x^2 + 4x + 5$:

• a. Is the graph concave up or concave down?

• b. Is the vertex the maximum or the minimum of the graph?

• c. Find the vertex of the parabola and give it as an ordered pair.

Background

Topic: Graphing Quadratic Functions

This question tests your understanding of the shape and features of a parabola, including concavity and the vertex.

Key Terms and Formulas:

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$, $y = f(x)$

• Concavity: If $a > 0$, the parabola opens up; if $a < 0$, it opens down.

Step-by-Step Guidance

1. Identify the value of $a$ in the quadratic function to determine concavity.

2. Use the sign of $a$ to decide if the vertex is a minimum or maximum.

3. Calculate the $x$-coordinate of the vertex using $x = -\frac{b}{2a}$.

4. Plug this $x$ value back into $f(x)$ to find the $y$-coordinate of the vertex.

Try solving on your own before revealing the answer!

Final Answer:

• a. Concave up

• b. Vertex is a minimum

• c. Vertex: $(-2, 1)$

The vertex formula and the sign of $a$ determine these properties.

Q8. For $f(x) = x^2 - 7x + 12$:

• a. Is the graph concave up or concave down?

• b. Is the vertex the maximum or the minimum of the graph?

• c. Find the vertex of the parabola and give it as an ordered pair.

Background

Topic: Graphing Quadratic Functions

This question tests your understanding of the shape and features of a parabola, including concavity and the vertex.

Key Terms and Formulas:

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$, $y = f(x)$

• Concavity: If $a > 0$, the parabola opens up; if $a < 0$, it opens down.

Step-by-Step Guidance

1. Identify the value of $a$ in the quadratic function to determine concavity.

2. Use the sign of $a$ to decide if the vertex is a minimum or maximum.

3. Calculate the $x$-coordinate of the vertex using $x = -\frac{b}{2a}$.

4. Plug this $x$ value back into $f(x)$ to find the $y$-coordinate of the vertex.

Try solving on your own before revealing the answer!

Final Answer:

• a. Concave up

• b. Vertex is a minimum

• c. Vertex: $(3.5, -0.25)$

The vertex formula and the sign of $a$ determine these properties.

Q9. For $f(x) = -(x + 3)^2 + 4$:

• a. Is the graph concave up or concave down?

• b. Is the vertex the maximum or the minimum of the graph?

• c. Find the vertex of the parabola and give it as an ordered pair.

Background

Topic: Graphing Quadratic Functions (Vertex Form)

This question tests your understanding of the vertex form of a quadratic and how to identify the vertex and concavity.

Key Terms and Formulas:

• Vertex Form: $f(x) = a(x - h)^2 + k$; vertex is $(h, k)$

• Concavity: If $a > 0$, opens up; if $a < 0$, opens down.

Step-by-Step Guidance

1. Identify $a$, $h$, and $k$ from the vertex form.

2. Determine concavity from the sign of $a$.

3. State whether the vertex is a maximum or minimum based on concavity.

4. Write the vertex as an ordered pair $(h, k)$, remembering to adjust the sign for $h$.

Try solving on your own before revealing the answer!

Final Answer:

• a. Concave down

• b. Vertex is a maximum

• c. Vertex: $(-3, 4)$

The vertex form makes it easy to identify these features.

Q10. For $f(x) = (x - 5)^2 - 2$:

• a. Is the graph concave up or concave down?

• b. Is the vertex the maximum or the minimum of the graph?

• c. Find the vertex of the parabola and give it as an ordered pair.

Background

Topic: Graphing Quadratic Functions (Vertex Form)

This question tests your understanding of the vertex form of a quadratic and how to identify the vertex and concavity.

Key Terms and Formulas:

• Vertex Form: $f(x) = a(x - h)^2 + k$; vertex is $(h, k)$

• Concavity: If $a > 0$, opens up; if $a < 0$, opens down.

Step-by-Step Guidance

1. Identify $a$, $h$, and $k$ from the vertex form.

2. Determine concavity from the sign of $a$.

3. State whether the vertex is a maximum or minimum based on concavity.

4. Write the vertex as an ordered pair $(h, k)$.

Try solving on your own before revealing the answer!

Final Answer:

• a. Concave up

• b. Vertex is a minimum

• c. Vertex: $(5, -2)$

The vertex form makes it easy to identify these features.