https://file.aiquickdraw.com/mathbot/file/1725415588833fo2y8qtv.jpg

It seems that the text extraction from the image wasn't successful. However, I can manually interpret and solve the equations from the image you provided. Let me analyze the equations visually and provide you with the answers.Here are the solutions for the equations from the image:

1. **\( x^2 + 4x = 21 \)**
   - Rearrange to form a standard quadratic equation: \( x^2 + 4x - 21 = 0 \).
   - Solve using the quadratic formula: 
     \[
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     \]
     where \( a = 1 \), \( b = 4 \), and \( c = -21 \).
   - \( x = \frac{-4 \pm \sqrt{16 + 84}}{2} \)
   - \( x = \frac{-4 \pm \sqrt{100}}{2} \)
   - \( x = \frac{-4 \pm 10}{2} \)
   - Solutions: \( x = 3 \) or \( x = -7 \).

2. **\( x^2 - 6x = 16 \)**
   - Rearrange to form a standard quadratic equation: \( x^2 - 6x - 16 = 0 \).
   - Solve using the quadratic formula:
     \[
     x = \frac{6 \pm \sqrt{36 + 64}}{2} 
     \]
   - \( x = \frac{6 \pm \sqrt{100}}{2} \)
   - Solutions: \( x = 8 \) or \( x = -2 \).

3. **\( x^2 + 3x = 6 \)**
   - Rearrange to form a standard quadratic equation: \( x^2 + 3x - 6 = 0 \).
   - Solve using the quadratic formula:
     \[
     x = \frac{-3 \pm \sqrt{9 + 24}}{2}
     \]
   - \( x = \frac{-3 \pm \sqrt{33}}{2} \)
   - Solutions: \( x = \frac{-3 + \sqrt{33}}{2} \) or \( x = \frac{-3 - \sqrt{33}}{2} \).

4. **\( x^2 - 9x = -8 \)**
   - Rearrange to form a standard quadratic equation: \( x^2 - 9x + 8 = 0 \).
   - Solve using the quadratic formula:
     \[
     x = \frac{9 \pm \sqrt{81 - 32}}{2}
     \]
   - \( x = \frac{9 \pm \sqrt{49}}{2} \)
   - Solutions: \( x = 8 \) or \( x = 1 \).

Would you like more details on any of these steps, or do you have any other questions? 

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Here are 5 additional questions to explore:

1. How can we apply the quadratic formula to equations in different forms?
2. What are the real-life applications of solving quadratic equations?
3. Can you solve a quadratic equation using factoring instead of the quadratic formula?
4. How does the discriminant in the quadratic formula determine the nature of the roots?
5. What is the importance of completing the square in solving quadratic equations?

**Tip:** Remember, when solving quadratic equations, always check if the equation can be factored easily before applying the quadratic formula. This can save time and effort!

Solve the following quadratic equations:
1. x^2 + 4x = 21
2. x^2 - 6x = 16
3. x^2 + 3x = 6
4. x^2 - 9x = -8

This webpage provides a step-by-step guide to solving quadratic equations such as x^2 + 4x = 21, x^2 - 6x = 16, and more using the quadratic formula. Explore key algebraic concepts and formulas, suitable for students in Grades 8-10.

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