C278 College Algebra

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Free C278 College Algebra Questions

1. What formula is used to calculate the perimeter of a rectangle?
  • P = 2L + 2W
  • P = L * W
  • P = 2(L + W)
  • P = L + W

Explanation

The perimeter of a rectangle is the total distance around the shape. A rectangle has two lengths (L) and two widths (W), so the perimeter can be calculated as the sum of all sides: P = L + W + L + W = 2L + 2W. This is equivalent to the formula P = 2(L + W), which is the standard formula used for calculating the perimeter efficiently.
2. If the quadratic equation 3x² − 5x + 2 = 0 is solved using the quadratic formula, what would be the first step in applying the formula?
  • Graph the quadratic function to find the roots.
  • Identify the coefficients a, b, and c from the equation.
  • Calculate the discriminant using the coefficients.
  • Factor the quadratic equation.

Explanation

The first step in using the quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), is to identify the coefficients a, b, and c from the quadratic equation in standard form ax² + bx + c = 0. For the equation 3x² − 5x + 2 = 0, the coefficients are a = 3, b = −5, and c = 2. Recognizing these coefficients is necessary before calculating the discriminant or applying the formula.
3.

Given the equation of a line y = mx + b, for what values of m will the line be decreasing? Enter the relation to 0.

  • m < 0

  • m > 0

  • m = 0

  • m ≥ 0

Explanation

Explanation

For a linear function y = mx + b, the slope m determines whether the line increases or decreases. A positive slope m > 0 rises as x increases, a zero slope (m = 0) is constant, and a negative slope (m < 0) falls as x increases. Therefore, the line is decreasing exactly when the slope is negative.

Correct Answer

m < 0


4. If the quadratic equation x² − 5x + 4 = 0 represents the area of a rectangle where the length is x and the width is (x − 4), what is the width when the area is zero?
  • 4
  • 1
  • 0
  • 5

Explanation

The area of a rectangle is given by length × width. Here, the length is x and the width is x − 4, so the area is A = x(x − 4). Setting the area to zero gives the equation x(x − 4) = 0. Solving this yields x = 0 or x = 4. Since the width is x − 4, substitute these values: for x = 0, width = 0 − 4 = −4 (not physically meaningful), and for x = 4, width = 4 − 4 = 0. Therefore, the width is zero when the area is zero.
5.

Consider the table of values below and find the linear function f(x) that corresponds to the table.

  • 4x − 3

  • 4x + 3

  • 3x − 4

  • 2x − 3

Explanation

Explanation

The outputs increase by 4 each step in x, so the slope is m=4. For a linear form f(x) = mx + b, plug in x = 0 ⇒ f(0)= −3 to get b = −3. Therefore, the function matching the table is f(x) = 4x − 3.

Correct Answer

4x − 3


6. Explain the process of solving the equation 5x − (4x − 1) = 2 in your own words.
  • First, distribute the negative sign, then combine like terms and isolate x.
  • First, multiply both sides by 5, then subtract 4x.
  • First, factor the equation, then solve for x.
  • First, add 1 to both sides, then solve for x.

Explanation

To solve 5x − (4x − 1) = 2, start by distributing the negative sign across the parentheses: 5x − 4x + 1 = 2. Combine like terms: (5x − 4x) + 1 = x + 1. Then, subtract 1 from both sides to isolate x: x + 1 − 1 = 2 − 1, giving x = 1. This step-by-step approach ensures that the variable is isolated properly while handling parentheses and combining terms correctly.
7. What is the degree of the polynomial resulting from the product of (x − 12)(x² + 9x − 3)?
  • 3
  • 2
  • 1
  • 4

Explanation

When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the factors. The first polynomial, x − 12, has degree 1, and the second polynomial, x² + 9x − 3, has degree 2. Therefore, the product will have degree 1 + 2 = 3.
8. What is the first step to solve the linear inequality 3x + 1
  • Add 1 to both sides.
  • Subtract 1 from both sides.
  • Divide both sides by 3.
  • Multiply both sides by 3.

Explanation

To solve the inequality 3x + 1 < 13, the first step is to isolate the variable term by removing the constant on the left side. This is done by subtracting 1 from both sides, giving 3x < 12. This step simplifies the inequality and prepares it for the next step of dividing by 3 to solve for x.
9. Describe the relationship between the first and second numbers based on the equation provided in the problem.
  • The first number is twice the second number.
  • The first number is always greater than the second number.
  • The sum of the two numbers is always positive.
  • Twice the second number equals 4 times the first number minus 2.

Explanation

The relationship between two numbers is determined by the equation that links them. If the equation shows that one number can be expressed as a multiple of the other, then that defines their relationship. For example, if the first number equals twice the second number, then for any value of the second number, the first number will always be exactly twice as large. This shows a direct proportional relationship between the first and second numbers.
10. Describe the significance of the discriminant in the quadratic formula.
  • The discriminant indicates the slope of the parabola.
  • The discriminant determines the nature of the roots of the quadratic equation.
  • The discriminant provides the vertex of the parabola.
  • The discriminant shows the maximum value of the quadratic function.

Explanation

The discriminant in the quadratic formula is the part under the square root: b² − 4ac. It determines the nature of the roots of a quadratic equation. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one repeated real root; and if it is negative, the roots are complex or imaginary. Thus, the discriminant gives critical information about whether solutions exist and what type they are.

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