Intermediate Algebra: Quadratics, Graphing, and Optimization Review

Q1. Solve using the square root property: $(7x + 2)^2 = 9$

Background

Topic: Solving Quadratic Equations by Square Root Property

This question tests your ability to solve a quadratic equation by isolating the squared term and then applying the square root property.

Key Terms and Formulas:

• Square Root Property: If $x^2 = a$, then $x = \pm\sqrt{a}$.

Step-by-Step Guidance

1. Start by isolating the squared term: $(7x + 2)^2 = 9$ is already isolated.

2. Take the square root of both sides: $7x + 2 = \pm\sqrt{9}$.

3. Simplify the square root: $\sqrt{9} = 3$, so $7x + 2 = \pm 3$.

4. Set up two equations: $7x + 2 = 3$ and $7x + 2 = -3$.

Try solving on your own before revealing the answer!

Q2. Determine the number and type of solutions: $2x^2 = 8x - 8$

Background

Topic: Discriminant and Nature of Solutions for Quadratic Equations

This question asks you to determine how many real or complex solutions a quadratic equation has by using the discriminant.

Key Terms and Formulas:

• Standard form: $ax^2 + bx + c = 0$

• Discriminant: $D = b^2 - 4ac$

• Interpretation:

  • If $D > 0$, two real solutions

  • If $D = 0$, one real solution

  • If $D < 0$, no real solutions (two complex)

Step-by-Step Guidance

1. Rewrite the equation in standard form: $2x^2 - 8x + 8 = 0$.

2. Identify $a = 2$, $b = -8$, $c = 8$.

3. Calculate the discriminant: $D = (-8)^2 - 4(2)(8)$.

Try solving on your own before revealing the answer!

Q3. Solve the equation using the quadratic formula: $x^2 = -6x + 2$

Background

Topic: Solving Quadratic Equations Using the Quadratic Formula

This question tests your ability to use the quadratic formula to solve a quadratic equation and express the answer in simplified radical form.

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: $x^2 + 6x - 2 = 0$.

2. Identify $a = 1$, $b = 6$, $c = -2$.

3. Plug these values into the quadratic formula.

4. Simplify under the square root: $b^2 - 4ac = 36 - 4(1)(-2)$.

Try solving on your own before revealing the answer!

Q4. Rationalize the denominator: $\frac{13}{\sqrt{19}}$

Background

Topic: Rationalizing Denominators

This question tests your ability to rewrite a fraction so that the denominator is a rational number.

Key Terms and Formulas:

• To rationalize $\frac{a}{\sqrt{b}}$, multiply numerator and denominator by $\sqrt{b}$.

Step-by-Step Guidance

1. Multiply numerator and denominator by $\sqrt{19}$: $\frac{13}{\sqrt{19}} \times \frac{\sqrt{19}}{\sqrt{19}}$.

2. Simplify the denominator: $\sqrt{19} \times \sqrt{19} = 19$.

3. Write the new numerator: $13 \sqrt{19}$.

Try solving on your own before revealing the answer!

Q5. Sketch the graph of $f(x) = (x + 4)^2 + 2$ and find the following:

• Domain

• Range

• $y$-intercept (if any)

• $x$-intercept (if any)

Background

Topic: Graphing Quadratic Functions and Identifying Key Features

This question tests your understanding of the vertex form of a quadratic function and how to determine its domain, range, and intercepts.

Key Terms and Formulas:

• Vertex form: $f(x) = a(x - h)^2 + k$

• Domain of any quadratic: all real numbers

• Range: depends on the vertex and direction of opening

• Intercepts: Set $x = 0$ for $y$-intercept, $f(x) = 0$ for $x$-intercepts

Step-by-Step Guidance

1. Identify the vertex: $(h, k) = (-4, 2)$.

2. Determine the direction the parabola opens (since $a = 1 > 0$, it opens upward).

3. State the domain: all real numbers.

4. State the range: $y \geq 2$.

5. Find the $y$-intercept by plugging $x = 0$ into $f(x)$.

6. Find the $x$-intercepts by setting $f(x) = 0$ and solving for $x$.

Try solving on your own before revealing the answer!

Q6. Find the value of $c$ for which $x^2 + 8x + c$ is a perfect-square trinomial. Then factor the perfect-square trinomial.

Background

Topic: Completing the Square and Factoring Perfect-Square Trinomials

This question tests your ability to recognize and create a perfect-square trinomial, then factor it.

Key Terms and Formulas:

• Perfect-square trinomial: $x^2 + 2bx + b^2 = (x + b)^2$

• To find $c$: $c = (\frac{8}{2})^2$

Step-by-Step Guidance

1. Identify $b$ in $x^2 + 8x + c$; here, $2b = 8 \implies b = 4$.

2. Calculate $c = b^2 = 4^2$.

3. Write the trinomial as $x^2 + 8x + 16$.

4. Factor as $(x + 4)^2$.

Try solving on your own before revealing the answer!

Q7. For the function $f(x) = x^2 - 8x + 15$:

• a) Find the vertex.

• b) Find the $x$-intercept(s) (if any).

• c) Find the $y$-intercept(s) (if any).

Background

Topic: Analyzing Quadratic Functions

This question tests your ability to find the vertex and intercepts of a quadratic function in standard form.

Key Terms and Formulas:

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

• $x$-intercepts: Set $f(x) = 0$ and solve for $x$

• $y$-intercept: Set $x = 0$ and solve for $f(0)$

Step-by-Step Guidance

1. Identify $a = 1$, $b = -8$, $c = 15$.

2. Find the vertex: $x = -\frac{-8}{2 \times 1}$.

3. Plug $x$ into $f(x)$ to find the $y$-coordinate of the vertex.

4. Find $x$-intercepts by solving $x^2 - 8x + 15 = 0$.

5. Find $y$-intercept by evaluating $f(0)$.

Try solving on your own before revealing the answer!

Q8. Simplify: $\sqrt{54}$ (Leave your answer in simplified radical form.)

Background

Topic: Simplifying Radicals

This question tests your ability to simplify square roots by factoring out perfect squares.

Key Terms and Formulas:

• Factor $54$into $9 \times 6$

• $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor $54$into $9 \times 6$.

2. Write $\sqrt{54} = \sqrt{9 \times 6}$.

3. Simplify $\sqrt{9} = 3$, so $\sqrt{54} = 3\sqrt{6}$.

Try solving on your own before revealing the answer!

Q9. You have 180 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What will the maximum area be? Include units in your answers.

Background

Topic: Optimization with Quadratic Functions

This question tests your ability to set up and solve an optimization problem involving area and perimeter constraints.

Key Terms and Formulas:

• Let $L$ = length (side parallel to river), $W$ = width (sides perpendicular to river)

• Perimeter constraint: $L + 2W = 180$

• Area: $A = L \times W$

Step-by-Step Guidance

1. Express $L$ in terms of $W$: $L = 180 - 2W$.

2. Write the area function: $A = (180 - 2W)W$.

3. Expand: $A = 180W - 2W^2$.

4. Find the value of $W$ that maximizes $A$ by finding the vertex of the parabola: $W = -\frac{b}{2a}$ for $A = -2W^2 + 180W$.

5. Plug this value of $W$ back into $L = 180 - 2W$ to find $L$.

6. Calculate the maximum area $A$ using these values.

Try solving on your own before revealing the answer!