Precalculus Study Guide: Quadratic, Radical, and Absolute Value Equations

Q1. When will the average cost of tuition and fees at public colleges reach a maximum?

Background

Topic: Quadratic Functions (Applications)

This question tests your understanding of how to find the maximum value of a quadratic function, which is important in modeling real-world scenarios.

Key formula:

The maximum (or minimum) of a quadratic function $T = -x^2 + 325x + 3101$ occurs at the vertex.

• Vertex formula for $ax^2 + bx + c$: $x = \frac{-b}{2a}$

Step-by-Step Guidance

1. Identify the coefficients: $a = -1$, $b = 325$, $c = 3101$.

2. Recall that the vertex of a parabola $ax^2 + bx + c$ is at $x = \frac{-b}{2a}$.

3. Set up the formula: $x = \frac{-325}{2 \times (-1)}$.

4. Calculate the denominator and numerator separately before plugging in the values.

Try solving on your own before revealing the answer!

Q2. Solve the radical equation: $\sqrt{2x + 5} + 5 = x$

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots and to check for extraneous solutions.

Key Terms and Formulas:

• Radical equation: An equation in which the variable is inside a root.

• Extraneous solution: A solution that does not satisfy the original equation after checking.

Step-by-Step Guidance

1. Isolate the radical: $\sqrt{2x + 5} = x - 5$.

2. Square both sides to eliminate the square root: $(\sqrt{2x + 5})^2 = (x - 5)^2$.

3. Expand the right side: $2x + 5 = x^2 - 10x + 25$.

4. Rearrange the equation to set it equal to zero: $x^2 - 12x + 20 = 0$.

Try solving on your own before revealing the answer!

Q3. Solve the quadratic equation: $2x^2 + 7x = 1$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by rearranging and using the quadratic formula.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Rewrite the equation in standard form: $2x^2 + 7x - 1 = 0$.

2. Identify coefficients: $a = 2$, $b = 7$, $c = -1$.

3. Set up the quadratic formula: $x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}$.

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

Try solving on your own before revealing the answer!

Q4. Solve the quadratic equation: $3x^2 + 3x - 3 = 0$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations using factoring or the quadratic formula.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify coefficients: $a = 3$, $b = 3$, $c = -3$.

2. Set up the quadratic formula: $x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3}$.

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

4. Plug the discriminant into the formula and simplify.

Try solving on your own before revealing the answer!

Q5. Solve the quadratic equation: $12x^2 + 24x = 12$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by rearranging and factoring.

Key Terms and Formulas:

• 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. Rewrite the equation in standard form: $12x^2 + 24x - 12 = 0$.

2. Divide both sides by 12 to simplify: $x^2 + 2x - 1 = 0$.

3. Identify coefficients: $a = 1$, $b = 2$, $c = -1$.

4. Set up the quadratic formula: $x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}$.

Try solving on your own before revealing the answer!

Q6. Solve the quadratic equation: $x^2 = 2x + 3$

Background

Topic: Quadratic Equations

This question tests your ability to rearrange and solve quadratic equations.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Rewrite the equation in standard form: $x^2 - 2x - 3 = 0$.

2. Identify coefficients: $a = 1$, $b = -2$, $c = -3$.

3. Set up the quadratic formula: $x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1}$.

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

Try solving on your own before revealing the answer!

Q7. Compute the discriminant, then solve the quadratic equation: $2x^2 + 7x - 14 = 0$

Background

Topic: Quadratic Equations (Discriminant)

This question tests your ability to compute the discriminant and use it to solve a quadratic equation.

Key Terms and Formulas:

• Discriminant: $D = b^2 - 4ac$

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

Step-by-Step Guidance

1. Identify coefficients: $a = 2$, $b = 7$, $c = -14$.

2. Compute the discriminant: $D = 7^2 - 4 \cdot 2 \cdot (-14)$.

3. Set up the quadratic formula: $x = \frac{-7 \pm \sqrt{D}}{2 \cdot 2}$.

4. Plug in the discriminant and simplify.

Try solving on your own before revealing the answer!

Q8. Solve the quadratic equation by factoring or completing the square: $x^2 = 5 - 4x$

Background

Topic: Quadratic Equations (Factoring/Completing the Square)

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

Key Terms and Formulas:

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

• Completing the square: Rewriting the equation to form a perfect square trinomial.

Step-by-Step Guidance

1. Rewrite the equation in standard form: $x^2 + 4x - 5 = 0$.

2. Try factoring: Look for two numbers that multiply to $-5$ and add to $4$.

3. If factoring is difficult, use completing the square: $x^2 + 4x = 5$.

4. Add $(\frac{4}{2})^2 = 4$ to both sides: $x^2 + 4x + 4 = 5 + 4$.

Try solving on your own before revealing the answer!

Q9. Consider the function $f(x) = 3x^2 - 4x - 8$

Background

Topic: Quadratic Functions (Graphing and Analysis)

This question tests your ability to analyze a quadratic function: finding the vertex, axis of symmetry, intercepts, domain, and range.

Key Terms and Formulas:

• Vertex: $x = \frac{-b}{2a}$

• Axis of symmetry: $x = \frac{-b}{2a}$

• Y-intercept: $f(0)$

• X-intercepts: Solve $f(x) = 0$

• Domain: All real numbers for quadratic functions

• Range: Depends on the direction of the parabola

Step-by-Step Guidance

1. Find the vertex: $x = \frac{-(-4)}{2 \cdot 3}$

2. Plug the $x$ value into $f(x)$ to get the $y$ coordinate of the vertex.

3. Axis of symmetry: $x = \frac{-b}{2a}$

4. Y-intercept: Evaluate $f(0)$.

5. X-intercepts: Set $f(x) = 0$ and solve for $x$ using the quadratic formula.

6. Domain: Quadratic functions have domain $(-\infty, \infty)$.

7. Range: Since $a > 0$, the parabola opens upward; range starts at the vertex $y$ value.

Try solving on your own before revealing the answer!

Q10. A ball is thrown upward and outward from a height of 7 feet. The height of the ball, $f(x)$ in feet, can be modeled by $f(x) = -0.1x^2 + 0.8x + 7$

Background

Topic: Quadratic Functions (Applications)

This question tests your ability to analyze a quadratic function in a real-world context: finding maximum height and horizontal distance.

Key Terms and Formulas:

• Vertex: $x = \frac{-b}{2a}$

• Maximum height: $f(x)$ at the vertex

• Horizontal distance before hitting the ground: Solve $f(x) = 0$

Step-by-Step Guidance

1. Find the vertex: $x = \frac{-0.8}{2 \cdot (-0.1)}$

2. Plug the $x$ value into $f(x)$ to find the maximum height.

3. To find how far the ball travels before hitting the ground, set $f(x) = 0$ and solve for $x$.

4. Use the quadratic formula for $-0.1x^2 + 0.8x + 7 = 0$.

Try solving on your own before revealing the answer!

Q11. Consider the function $h(x) = |x - 5| - 8$

Background

Topic: Absolute Value Functions

This question tests your ability to analyze absolute value functions: finding intercepts, evaluating at a point, and determining the range.

Key Terms and Formulas:

• Absolute value function: $h(x) = |x - 5| - 8$

• Y-intercept: $h(0)$

• X-intercepts: Solve $|x - 5| - 8 = 0$

• Range: For $h(x) = |x - 5| - 8$, minimum value is $-8$

Step-by-Step Guidance

1. Find the Y-intercept: Evaluate $h(0)$.

2. Find the X-intercepts: Set $|x - 5| - 8 = 0$ and solve for $x$.

3. Find $h(x)$ when $x = 2$: Substitute $x = 2$ into the function.

4. Determine the range: The minimum value is $-8$, and the function increases without bound.

Try solving on your own before revealing the answer!