To solve the quadratic equation \( 2x^2 + 17x + 21 = 0 \), we will use the quadratic formula:

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

For the given equation \( 2x^2 + 17x + 21 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 17 \)
- \( c = 21 \)

Now, substitute the values into the quadratic formula:

\[
x = \frac{-17 \pm \sqrt{17^2 - 4(2)(21)}}{2(2)}
\]

Let's calculate it step by step:

1. Calculate the discriminant: \( \Delta = b^2 - 4ac \)

\[
\Delta = 17^2 - 4(2)(21) = 289 - 168 = 121
\]

2. Now, substitute the discriminant back into the formula:

\[
x = \frac{-17 \pm \sqrt{121}}{4}
\]

\[
x = \frac{-17 \pm 11}{4}
\]

3. Solve for the two possible values of \( x \):

\[
x_1 = \frac{-17 + 11}{4} = \frac{-6}{4} = -\frac{3}{2}
\]

\[
x_2 = \frac{-17 - 11}{4} = \frac{-28}{4} = -7
\]

So, the solutions are:

\[
x = -\frac{3}{2} \quad \text{or} \quad x = -7
\]

Would you like further clarification on any of the steps?

Here are 5 related questions:
1. How can you verify the solutions to a quadratic equation?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you complete the square to solve a quadratic equation?
4. Can all quadratic equations be factored without using the quadratic formula?
5. What are the real-life applications of solving quadratic equations?

**Tip:** The sign of the discriminant (\( \Delta \)) tells you about the nature of the roots. If \( \Delta > 0 \), there are two distinct real roots. If \( \Delta = 0 \), there is one real root. If \( \Delta < 0 \), the roots are complex.

Learn how to solve the quadratic equation 2x^2 + 17x + 21 using the quadratic formula. This detailed solution includes calculating the discriminant and determining the nature of the roots. Perfect for students in Grades 8-10.

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