https://file.aiquickdraw.com/mathbot/file/1724843498167qrmlvq7k.jpg

I can see that the image contains multiple equations. Let me help by focusing on solving the quadratic equations from the image. I'll break down the process for each relevant one.

### 1. Quadratic equation: \( x^2 + 8x = 9 \)
Rearranging to get the equation in standard quadratic form:
\[
x^2 + 8x - 9 = 0
\]
Now, solving this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( x^2 + 8x - 9 = 0 \), we have:
- \( a = 1 \)
- \( b = 8 \)
- \( c = -9 \)

Substituting these values into the quadratic formula:
\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)}
\]
\[
x = \frac{-8 \pm \sqrt{64 + 36}}{2}
\]
\[
x = \frac{-8 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-8 \pm 10}{2}
\]
So the two solutions are:
\[
x = \frac{-8 + 10}{2} = 1 \quad \text{or} \quad x = \frac{-8 - 10}{2} = -9
\]
Thus, the solutions are:
\[
x = 1 \quad \text{or} \quad x = -9
\]

### 2. Quadratic equation: \( x^2 + 8x + 9 = 0 \)
Here, we solve directly using the quadratic formula again:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( x^2 + 8x + 9 = 0 \), we have:
- \( a = 1 \)
- \( b = 8 \)
- \( c = 9 \)

Substituting into the formula:
\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(9)}}{2(1)}
\]
\[
x = \frac{-8 \pm \sqrt{64 - 36}}{2}
\]
\[
x = \frac{-8 \pm \sqrt{28}}{2}
\]
\[
x = \frac{-8 \pm 2\sqrt{7}}{2}
\]
Simplifying:
\[
x = -4 \pm \sqrt{7}
\]
Thus, the solutions are:
\[
x = -4 + \sqrt{7} \quad \text{or} \quad x = -4 - \sqrt{7}
\]

Would you like more details on any of these steps, or do you have any other questions? Here are 5 related questions you might want to explore:

1. How do you factor quadratic equations manually?
2. Can the quadratic formula be used for non-quadratic equations?
3. What are the conditions for a quadratic equation to have real, imaginary, or equal roots?
4. How does the discriminant affect the type of solutions in quadratic equations?
5. What is the connection between the graph of a quadratic function and its roots?

**Tip:** If the discriminant \( b^2 - 4ac \) is positive, the equation will have two real roots; if it is zero, there will be exactly one real root; and if negative, the roots will be complex (imaginary).

Learn how to solve the quadratic equation x^2 + 8x = 9 step-by-step with detailed explanations and examples. Understand the quadratic formula and how it applies to solving equations like x^2 + 8x = 9. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation