Solving a Quadratic Equation Using the Quadratic Formula

Q1. Solve for $t$ in the equation $3t^2 + 18t + 16 = 10$. Express your answer in radical form. If there is more than one answer, enter all answers, separated by commas. If no real solution exists, enter DNE.

Background

Topic: Quadratic Equations

This question tests your ability to solve a quadratic equation by rearranging it into standard form and applying the quadratic formula. You are also asked to express your answer in radical form, which means you should not simplify the square root to a decimal.

Key Terms and Formulas

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.

• Quadratic Formula:

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

• $a$, $b$, and $c$ are the coefficients from the standard form of the quadratic equation.

• The expression under the square root, $b^2 - 4ac$, is called the discriminant.

Step-by-Step Guidance

1. First, rewrite the equation in standard quadratic form ($ax^2 + bx + c = 0$) by subtracting 10 from both sides:

   $3t^2 + 18t + 16 - 10 = 0$

   Simplify the constants.

2. Identify the coefficients $a$, $b$, and $c$ from your simplified equation.

3. Write down the quadratic formula:

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

4. Substitute the values of $a$, $b$, and $c$ into the quadratic formula. Carefully compute the discriminant $b^2 - 4ac$.

5. Set up the two possible solutions for $t$ using the $\pm$ symbol, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Final Answer:

$t = \frac{ -18 \pm \sqrt{100} }{6}$

There are two real solutions because the discriminant is positive. Substitute and simplify to get the two values for $t$ in radical form.