https://file.aiquickdraw.com/mathbot/file/172477246940063aoox5o.jpg

I'll solve the math problems visible in the image. Let's start with the first few problems.

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### Problem 1: \( x^2 - 3x + 2 = 0 \)

This is a quadratic equation. We can solve it using the quadratic formula:

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

Here, \( a = 1 \), \( b = -3 \), and \( c = 2 \).

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(2)}}{2(1)}
\]

\[
x = \frac{3 \pm \sqrt{9 - 8}}{2}
\]

\[
x = \frac{3 \pm \sqrt{1}}{2}
\]

\[
x = \frac{3 \pm 1}{2}
\]

So, \( x = 2 \) or \( x = 1 \).

---

### Problem 2: \( x^2 + x - 6 = 0 \)

Again, using the quadratic formula:

Here, \( a = 1 \), \( b = 1 \), and \( c = -6 \).

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-6)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 24}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{25}}{2}
\]

\[
x = \frac{-1 \pm 5}{2}
\]

So, \( x = 2 \) or \( x = -3 \).

---

### Problem 3: \( 2x^2 - 3x - 5 = 0 \)

For this quadratic equation:

Here, \( a = 2 \), \( b = -3 \), and \( c = -5 \).

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)}
\]

\[
x = \frac{3 \pm \sqrt{9 + 40}}{4}
\]

\[
x = \frac{3 \pm \sqrt{49}}{4}
\]

\[
x = \frac{3 \pm 7}{4}
\]

So, \( x = \frac{10}{4} = 2.5 \) or \( x = \frac{-4}{4} = -1 \).

---

### Problem 4: \( x^2 - 5x + 6 = 0 \)

Using the quadratic formula:

Here, \( a = 1 \), \( b = -5 \), and \( c = 6 \).

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}
\]

\[
x = \frac{5 \pm \sqrt{25 - 24}}{2}
\]

\[
x = \frac{5 \pm \sqrt{1}}{2}
\]

\[
x = \frac{5 \pm 1}{2}
\]

So, \( x = 3 \) or \( x = 2 \).

---

### Problem 5: \( x^2 - 2x - 8 = 0 \)

For this equation:

Here, \( a = 1 \), \( b = -2 \), and \( c = -8 \).

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 32}}{2}
\]

\[
x = \frac{2 \pm \sqrt{36}}{2}
\]

\[
x = \frac{2 \pm 6}{2}
\]

So, \( x = 4 \) or \( x = -2 \).

---

Would you like further details or have any questions? 

Here are some related questions:

1. How would you solve quadratic equations by completing the square?
2. Can you explain the discriminant and how it determines the nature of the roots?
3. How do you factorize quadratic equations when the coefficient of \( x^2 \) is not 1?
4. What is the significance of the vertex in the graph of a quadratic function?
5. How do you derive the quadratic formula?

**Tip:** Always check the discriminant \( b^2 - 4ac \) before solving a quadratic equation, as it gives you valuable information about the roots.

Learn how to solve multiple quadratic equations: x^2 - 3x + 2, x^2 + x - 6, 2x^2 - 3x - 5, x^2 - 5x + 6, and x^2 - 2x - 8 with step-by-step solutions using the quadratic formula. This page covers key algebraic concepts and formulas for solving quadratic equations. Suitable for students in Grades 9-12.

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