Question 1

Which of the following equations is an example of a rational equation?

Accepted Answer

$$\frac{1}{x} + \frac{12}{5x} = 10$$

Question 2

Given the equation $$-2x + 6(x + 1) = -3x - 1$$, what is the solution for $$x$$?

Accepted Answer

$$x = 1$$

Question 3

If a quadratic equation $$ax^2 + bx + c = 0$ has a discriminant D$$ = $$b^2 - 4ac$$ \u003c 0$$, what can you conclude about its solutions?

Accepted Answer

There are no real solutions.

Question 4

Which of the following best describes the solution set for the equation $$2x + 2 = 2x - 4$$?

Accepted Answer

The solution set is the empty set.

Question 5

A chemical reaction is modeled by the equation $$x^2 - 10x + 21 = 0$, where x$ is the concentration of a reactant. What are the possible concentrations?

Accepted Answer

$$x = 3$$ and $$x = 7$$

Question 6

Which of the following is the correct first step when solving the quadratic equation $$x^2 - 5x = 4$?

Accepted Answer

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

Question 7

A medical dosage calculation leads to the equation $$\sqrt{6x + 4} + 2 = x. $$What is the first step to isolate $$x$$?

Accepted Answer

Subtract 2 from both sides.

Question 8

If the equation $$15y - 3 = 15y - 3 is $$given, what is the solution set?

Accepted Answer

All real numbers

Question 9

Which of the following statements about the discriminant $$D = b^2$ - 4ac is $$true?

Accepted Answer

If $$D \u003c 0$$, there are two complex solutions.

Question 10

A quadratic equation $$x^2 - 10x + 21 = 0$ is solved by factoring. Which pair of numbers is used to factor the equation?

Accepted Answer

$$3$$ and $$7$$

Question 11

Which of the following is NOT a possible outcome when solving an equation?

Accepted Answer

The equation has exactly three real solutions.

Question 12

A rational equation is given by $$\frac{1}{x} + \frac{12}{5x} = 10. $$What is the least common denominator (LCD) you should use to clear the denominators?

Accepted Answer

$$5x$$

Question 13

If you solve $$x^2 - 10x + 21 = 0$ by factoring, what should you do after finding the potential solutions?

Accepted Answer

Check the solutions in the original equation.

Question 14

Which of the following best describes the process of solving a linear equation?

Accepted Answer

Isolate all terms involving $$x on $$one side and constants on the other.

Question 15

A quadratic equation $$ax^2 + bx + c = 0$ is solved using the quadratic formula. What is the formula?

Accepted Answer

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

Question 16

In a real-world scenario, a company’s profit is modeled by $$x^2 - 5x = 4$. What is the first step to find the break-even points?

Accepted Answer

Rewrite as $$x^2 - 5x - 4 = 0$

Question 17

If you have the equation $$x^2 - 10x + 21 = 0$ and you factor it as (x - 3)(x - 7) = 0$, what are the solutions?

Accepted Answer

$$x = 3$$ and $$x = 7$$

Question 18

Which of the following is the correct interpretation of the symbol $$\iff$$?

Accepted Answer

A is equivalent to B (A if and only if B)

Question 19

If you solve a quadratic equation and the discriminant $$D = 0$$, what does this mean?

Accepted Answer

There is one real solution (a repeated root).

Solving Equations: Linear, Quadratic, and Beyond

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