Q: What is the discriminant for $$3t^2 + 18t + 6 = 0$?

$$18^2 - 4$$ \times 3 \times 6$$

Question 1

What is the first step to solve the quadratic equation $$3t^2 + 18t + 16 = 10$ for t$?

Accepted Answer

Subtract 10 from both sides to set the equation to zero.

Question 2

After simplifying $$3t^2 + 18t + 16 = 10$, what is the resulting equation?

Accepted Answer

$$3t^2 + 18t + 6 = 0$

Question 3

Which formula should be used to solve $$3t^2 + 18t + 6 = 0$?

Accepted Answer

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

Question 4

For the equation $$3t^2 + 18t + 6 = 0$, what are the values of a$, b$, and c$ in the quadratic formula?

Accepted Answer

$$a = 3, b = 18, c = 6$$

Question 5

What is the discriminant for $$3t^2 + 18t + 6 = 0$?

Accepted Answer

$$18^2 - 4$$ 	imes 3 	imes 6$$

Question 6

Calculate the value of the discriminant for $$3t^2 + 18t + 6 = 0$.

Accepted Answer

$$252$$

Question 7

Express the solutions for $$t in $$radical form for $$3t^2 + 18t + 6 = 0$.

Accepted Answer

$$t = \frac{-18 \pm \sqrt{252}}{6}$$

Question 8

Simplify $$\sqrt{252} in $$the solution $$t = \frac{-18 \pm \sqrt{252}}{6}.$$

Accepted Answer

$$6\sqrt{7}$$

Question 9

What are the final solutions for $$t in $$simplest radical form for $$3t^2 + 18t + 6 = 0$?

Accepted Answer

$$t = \frac{-3 \pm \sqrt{7}}{1}$$

Question 10

If a quadratic equation has a positive discriminant, what does this indicate about its solutions?

Accepted Answer

There are two distinct real solutions.

Question 11

Which of the following best describes the process of solving $$3t^2 + 18t + 16 = 10$ in a real-world scenario, such as calculating the time for a projectile to reach a certain height?

Accepted Answer

Set the equation equal to zero, use the quadratic formula, and interpret the solutions in context.

Question 12

If the quadratic equation $$3t^2 + 18t + 6 = 0$ models the concentration of a chemical over time, what does each solution for t$ represent?

Accepted Answer

The time when the concentration reaches a specific value.

Question 13

Which step is necessary to check if the solutions to $$3t^2 + 18t + 6 = 0$ are valid in a real-world context?

Accepted Answer

Verify that the solutions are non-negative if negative time is not meaningful.

Question 14

If $$t$$ represents time in a medical dosage equation $$3t^2 + 18t + 6 = 0$, what should you do if one solution is negative?

Accepted Answer

Discard the negative solution as it is not physically meaningful.

Question 15

Which of the following is a common mistake when applying the quadratic formula?

Accepted Answer

Forgetting to divide by $$2a in $$the formula.

Question 16

If the equation $$3t^2 + 18t + 6 = 0$ is used to predict the time when a population reaches a certain size, what does the discriminant tell you?

Accepted Answer

Whether the population will reach that size at one, two, or no times.

Question 17

If you are given $$3t^2 + 18t + 16 = 10$ and asked to solve for t$ in radical form, which answer is correct?

Accepted Answer

$$t = \frac{-18 \pm \sqrt{252}}{6}$$

Solving a Quadratic Equation Using the Quadratic Formula

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