College Algebra Midterm Study Guidance (MATH 206)

Q1. Simplify the expression: $-4\sqrt{20} + \sqrt{80} + \sqrt{45}$

Background

Topic: Simplifying Radical Expressions

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

Key Terms and Formulas

• Square root: $\sqrt{a}$ is the non-negative number whose square is $a$.

• Simplifying radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

• Look for perfect square factors (like 4, 9, 16, 25, etc.) inside the radicand.

Step-by-Step Guidance

1. Break down each radicand (number under the square root) into its prime factors to find perfect squares.

2. Simplify each square root by taking out the square root of the perfect square factor.

3. Multiply the coefficients outside the radicals by the simplified values.

4. Combine like terms (terms with the same radical part).

Try solving on your own before revealing the answer!

Final Answer: $-2\sqrt{5}$

Each radical simplifies as follows:

$-4\sqrt{20} = -4 \times 2\sqrt{5} = -8\sqrt{5}$

$\sqrt{80} = 4\sqrt{5}$

$\sqrt{45} = 3\sqrt{5}$

Combine: $-8\sqrt{5} + 4\sqrt{5} + 3\sqrt{5} = -8 + 4 + 3 = -1$ times $\sqrt{5}$, so $-\sqrt{5}$.

But double-check the coefficients: $-8\sqrt{5} + 4\sqrt{5} + 3\sqrt{5} = -1\sqrt{5}$, so the answer is $-\sqrt{5}$, which matches option C.

Q2. Simplify the expression: $(\sqrt{5} - 4)^2$

Background

Topic: Expanding Binomials with Radicals

This question tests your ability to expand and simplify binomials involving square roots.

Key Terms and Formulas

• Binomial square: $(a - b)^2 = a^2 - 2ab + b^2$

• Square root properties: $(\sqrt{a})^2 = a$

Step-by-Step Guidance

1. Identify $a = \sqrt{5}$ and $b = 4$ in the binomial.

2. Apply the binomial square formula: $(a - b)^2 = a^2 - 2ab + b^2$.

3. Calculate $a^2$, $2ab$, and $b^2$ separately.

4. Combine the results to write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer: $21 - 8\sqrt{5}$

Expanding: $(\sqrt{5} - 4)^2 = (\sqrt{5})^2 - 2 \times \sqrt{5} \times 4 + 4^2 = 5 - 8\sqrt{5} + 16 = 21 - 8\sqrt{5}$.

Q3. Simplify the expression: $(2x^3 - 3x^2 + 4x + 3) - x(7x^2 - 6x + 2)$

Background

Topic: Polynomial Operations

This question tests your ability to distribute and combine like terms in polynomial expressions.

Key Terms and Formulas

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

• Combining like terms: Add or subtract coefficients of terms with the same degree.

Step-by-Step Guidance

1. Distribute $x$ across the second polynomial: $x(7x^2 - 6x + 2)$.

2. Subtract the result from the first polynomial, being careful with signs.

3. Combine like terms for each degree of $x$.

4. Write the simplified polynomial in standard form (descending powers of $x$).

Try solving on your own before revealing the answer!

Final Answer: $-5x^3 + 3x^2 + 2x + 3$

After distributing and combining like terms, the simplified expression is $-5x^3 + 3x^2 + 2x + 3$.

Q4. Rationalize the denominator: $\frac{\sqrt{7}}{2 - \sqrt{7}}$

Background

Topic: Rationalizing Denominators

This question tests your ability to eliminate radicals from the denominator by multiplying by the conjugate.

Key Terms and Formulas

• Conjugate: For $a - b$, the conjugate is $a + b$.

• Multiplying by the conjugate: $\frac{1}{a - b} \times \frac{a + b}{a + b}$

• Difference of squares: $(a - b)(a + b) = a^2 - b^2$

Step-by-Step Guidance

1. Identify the conjugate of the denominator: $2 + \sqrt{7}$.

2. Multiply numerator and denominator by the conjugate.

3. Apply the difference of squares formula to the denominator.

4. Simplify the numerator and denominator as much as possible.

Try solving on your own before revealing the answer!

Final Answer: $\frac{7 + 2\sqrt{7}}{-3}$

Multiplying numerator and denominator by $2 + \sqrt{7}$ gives $\frac{\sqrt{7}(2 + \sqrt{7})}{(2 - \sqrt{7})(2 + \sqrt{7})} = \frac{2\sqrt{7} + 7}{4 - 7} = \frac{2\sqrt{7} + 7}{-3}$.

Q5. Rationalize the denominator: $\frac{\sqrt{3}}{\sqrt{2} + 3\sqrt{3}}$

Background

Topic: Rationalizing Denominators with Binomials

This question tests your ability to rationalize denominators that are binomials with radicals.

Key Terms and Formulas

• Conjugate: For $a + b$, the conjugate is $a - b$.

• Multiply numerator and denominator by the conjugate of the denominator.

• Difference of squares: $(a + b)(a - b) = a^2 - b^2$

Step-by-Step Guidance

1. Identify the conjugate of the denominator: $\sqrt{2} - 3\sqrt{3}$.

2. Multiply numerator and denominator by this conjugate.

3. Expand both numerator and denominator using distributive property and difference of squares.

4. Simplify the resulting expression as much as possible.

Try solving on your own before revealing the answer!

Final Answer: $\frac{9 - \sqrt{6}}{25}$

After multiplying by the conjugate and simplifying, the result is $\frac{9 - \sqrt{6}}{25}$.

Q6. Simplify the expression: $\frac{2x + 10}{x^2 - 25}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials and reduce rational expressions.

Key Terms and Formulas

• Factoring: $x^2 - 25$ is a difference of squares.

• Cancel common factors in numerator and denominator.

Step-by-Step Guidance

1. Factor the denominator as a difference of squares: $x^2 - 25 = (x - 5)(x + 5)$.

2. Factor the numerator: $2x + 10 = 2(x + 5)$.

3. Write the expression with the factored numerator and denominator.

4. Identify and cancel any common factors.

Try solving on your own before revealing the answer!

Final Answer: $\frac{2}{x - 5}$

After factoring and canceling $(x + 5)$, the simplified expression is $\frac{2}{x - 5}$.

Q7. Factor the polynomial completely: $x^2 - 4x - 32$

Background

Topic: Factoring Quadratic Polynomials

This question tests your ability to factor quadratics of the form $x^2 + bx + c$.

Key Terms and Formulas

• Quadratic factoring: Find two numbers that multiply to $c$ and add to $b$.

• Standard form: $x^2 + bx + c$

Step-by-Step Guidance

1. Identify $b = -4$ and $c = -32$.

2. Find two numbers whose product is $-32$ and whose sum is $-4$.

3. Write the factors as $(x + m)(x + n)$, where $m$ and $n$ are the numbers found.

Try solving on your own before revealing the answer!

Final Answer: $(x - 8)(x + 4)$

The numbers $-8$ and $4$multiply to$-32$and add to$-4$, so the factorization is$(x - 8)(x + 4)$.

Q8. Factor the polynomial completely: $3x^2 + 10x + 3$

Background

Topic: Factoring Quadratics with Leading Coefficient Not 1

This question tests your ability to factor quadratics where the leading coefficient is not 1.

Key Terms and Formulas

• Factoring by grouping or trial: Find two numbers that multiply to $a \times c$ and add to $b$.

• Standard form: $ax^2 + bx + c$

Step-by-Step Guidance

1. Identify $a = 3$, $b = 10$, $c = 3$.

2. Find two numbers that multiply to $3 \times 3 = 9$ and add to $10$.

3. Rewrite the middle term using these numbers and factor by grouping.

4. Write the final factorization.

Try solving on your own before revealing the answer!

Final Answer: $(3x + 1)(x + 3)$

The numbers $1$and $9$ multiply to $9$and add to $10$, so the factorization is $(3x + 1)(x + 3)$.

Q9. Solve the equation: $x(2x - 5) = (2x + 2)(x - 3)$

Background

Topic: Solving Quadratic Equations

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

Key Terms and Formulas

• Expand both sides of the equation.

• Move all terms to one side to set the equation to zero.

• Factor or use the quadratic formula to solve for $x$.

Step-by-Step Guidance

1. Expand both sides: $x(2x - 5)$ and $(2x + 2)(x - 3)$.

2. Bring all terms to one side to form a quadratic equation equal to zero.

3. Combine like terms and write the equation in standard quadratic form.

4. Factor or use the quadratic formula to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 6$

After expanding and simplifying, the solution is $x = 6$.

Q10. Solve the equation: $x^2 + 5x + 6 = 0$

Background

Topic: Solving Quadratic Equations by Factoring

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

Key Terms and Formulas

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

• Factoring: Find two numbers that multiply to $c$ and add to $b$.

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

Step-by-Step Guidance

1. Identify $b = 5$ and $c = 6$.

2. Find two numbers that multiply to $6$and add to $5$.

3. Write the factors as $(x + m)(x + n) = 0$.

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

Try solving on your own before revealing the answer!

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

The factors are $(x + 2)(x + 3) = 0$, so $x = -2$ or $x = -3$.

Q11. Solve the equation: $\sqrt{2x - 4} = 4$

Background

Topic: Solving Radical Equations

This question tests your ability to solve equations involving square roots.

Key Terms and Formulas

• To eliminate the square root, square both sides of the equation.

• Isolate the variable after removing the radical.

Step-by-Step Guidance

1. Square both sides to eliminate the square root.

2. Solve the resulting linear equation for $x$.

3. Check your solution by substituting back into the original equation to ensure it is valid.

Try solving on your own before revealing the answer!

Final Answer: $x = 10$

Squaring both sides gives $2x - 4 = 16$, so $x = 10$.

Q12. Solve the inequality: $2x - 5 > 3$

Background

Topic: Linear Inequalities

This question tests your ability to solve and interpret linear inequalities.

Key Terms and Formulas

• Isolate $x$ by performing inverse operations.

• Remember to reverse the inequality sign if you multiply or divide by a negative number (not needed here).

Step-by-Step Guidance

1. Add $5$to both sides to isolate the $2x$ term.

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

3. Write the solution in inequality form.

Try solving on your own before revealing the answer!

Final Answer: $x > 4$

Adding $5$gives $2x > 8$, so $x > 4$.

Q13. Solve the inequality: $-15 \leq 9 - 4x \leq 25$

Background

Topic: Compound Linear Inequalities

This question tests your ability to solve compound inequalities and express the solution as an interval.

Key Terms and Formulas

• Isolate the variable in the middle by performing the same operation on all three parts.

• Reverse the inequality sign if you multiply or divide by a negative number.

Step-by-Step Guidance

1. Subtract $9$ from all three parts of the inequality.

2. Divide all parts by $-4$, remembering to reverse the inequality signs.

3. Write the solution in the form $a \leq x \leq b$.

Try solving on your own before revealing the answer!

Final Answer: $-4 \leq x \leq 6$

After solving, the solution is $-4 \leq x \leq 6$.

Q14. Write an equation for a line passing through $(1, 3)$ and whose slope is $2$.

Background

Topic: Equation of a Line (Point-Slope and Slope-Intercept Form)

This question tests your ability to write the equation of a line given a point and a slope.

Key Terms and Formulas

• Point-slope form: $y - y_1 = m(x - x_1)$

• Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

1. Identify the point $(x_1, y_1) = (1, 3)$ and the slope $m = 2$.

2. Plug these values into the point-slope form.

3. Simplify the equation to slope-intercept form.

Try solving on your own before revealing the answer!

Final Answer: $y = 2x + 1$

Using point-slope form and simplifying, the equation is $y = 2x + 1$.

Q15. Write the equation of a circle with center at $(-3, 2)$ and radius $4$.

Background

Topic: Equation of a Circle

This question tests your ability to write the equation of a circle given its center and radius.

Key Terms and Formulas

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

Step-by-Step Guidance

1. Identify the center $(h, k) = (-3, 2)$ and radius $r = 4$.

2. Plug these values into the standard form of the circle equation.

3. Simplify the equation as needed.

Try solving on your own before revealing the answer!

Final Answer: $(x + 3)^2 + (y - 2)^2 = 16$

Plugging in the center and radius, the equation is $(x + 3)^2 + (y - 2)^2 = 16$.

Q16(a). Find the domain of the function $f(x) = \sqrt{x - 2}$ (do not graph).

Background

Topic: Domain of Functions Involving Square Roots

This question tests your ability to determine the set of $x$-values for which a function involving a square root is defined.

Key Terms and Formulas

• The expression under a square root must be greater than or equal to zero for real-valued functions.

Step-by-Step Guidance

1. Set the radicand (expression under the square root) greater than or equal to zero: $x - 2 \geq 0$.

2. Solve the inequality for $x$.

3. Express the domain in interval notation.

Try solving on your own before revealing the answer!

Final Answer: $[2, \infty)$

The function is defined for $x \geq 2$, so the domain is $[2, \infty)$.

Q16(b). Find the domain of the function $g(x) = \frac{5}{x + 4}$ (do not graph).

Background

Topic: Domain of Rational Functions

This question tests your ability to determine the set of $x$-values for which a rational function is defined.

Key Terms and Formulas

• The denominator of a rational function cannot be zero.

Step-by-Step Guidance

1. Set the denominator not equal to zero: $x + 4 \neq 0$.

2. Solve for $x$ to find the excluded value.

3. Express the domain in interval notation, excluding the value found.

Try solving on your own before revealing the answer!

Final Answer: $(-\infty, -4) \cup (-4, \infty)$

The function is defined for all $x$ except $x = -4$.

Q17(a). The length of a rectangle is 3 meters less than twice the width. If the area of the rectangle is 665 square meters, find the dimensions (the width and the length).

Background

Topic: Quadratic Word Problems (Geometry)

This question tests your ability to translate a word problem into an equation, form a quadratic, and solve for the unknowns.

Key Terms and Formulas

• Area of a rectangle: $A = \text{length} \times \text{width}$

• Let $w$ be the width, then length is $2w - 3$.

Step-by-Step Guidance

1. Let $w$ be the width. Express the length in terms of $w$.

2. Write the area equation: $w \times (2w - 3) = 665$.

3. Expand and rearrange the equation to standard quadratic form.

4. Solve the quadratic equation for $w$ (width).

5. Once you have $w$, substitute back to find the length.

Try solving on your own before revealing the answer!

Final Answer: Width = 19 m, Length = 35 m

Solving $w(2w - 3) = 665$ gives $w = 19$, so length is $2 \times 19 - 3 = 35$.

Q18(a). Sketch the graph of the function $f(x) = (x + 1)^2 - 2$, starting from the graph of $g(x) = x^2$.

Background

Topic: Graph Transformations

This question tests your understanding of how to apply horizontal and vertical shifts to the graph of a basic quadratic function.

Key Terms and Formulas

• Parent function: $g(x) = x^2$

• Horizontal shift: $f(x) = (x + h)^2$ shifts left by $h$ units if $h > 0$.

• Vertical shift: $f(x) = x^2 + k$ shifts up by $k$ units if $k > 0$, down if $k < 0$.

Step-by-Step Guidance

1. Identify the parent function: $g(x) = x^2$.

2. Recognize the horizontal shift: $(x + 1)^2$ means shift left by 1 unit.

3. Recognize the vertical shift: $-2$ means shift down by 2 units.

4. Describe the new vertex and the direction of the parabola.

5. Sketch the graph using these transformations.

Try sketching on your own before revealing the answer!

Final Answer:

The graph of $f(x) = (x + 1)^2 - 2$ is a parabola opening upwards with vertex at $(-1, -2)$, shifted left 1 unit and down 2 units from $g(x) = x^2$.