https://file.aiquickdraw.com/mathbot/file/1726624519073jowxshev.jpg

I see several quadratic equations in the image you've uploaded. I'll list and solve them one by one.

### 1. Equation 1: 
\[
x^2 - 5x + 9 = 0
\]

This is a standard quadratic equation, which we can solve using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -5 \), and \( c = 9 \).

Let's calculate this.

### 2. Equation 2:
\[
4x^2 + 8x - 60 = 0
\]
This is also a quadratic equation. Here \( a = 4 \), \( b = 8 \), and \( c = -60 \). We can solve it using the quadratic formula or factoring.

### 3. Equation 3:
\[
2x^2 - 2 - 3x = 0
\]
Rearranging it in standard form:
\[
2x^2 - 3x - 2 = 0
\]
Now, \( a = 2 \), \( b = -3 \), and \( c = -2 \).

### 4. Equation 4:
\[
36x^2 + 25 = 60x
\]
Rearranging:
\[
36x^2 - 60x + 25 = 0
\]
Here \( a = 36 \), \( b = -60 \), and \( c = 25 \).

Let me solve these equations step-by-step using the quadratic formula.Here are the solutions for each equation:

### 1. \( x^2 - 5x + 9 = 0 \):
This equation has complex solutions:
\[
x_1 = 2.5 + 1.66i, \quad x_2 = 2.5 - 1.66i
\]

### 2. \( 4x^2 + 8x - 60 = 0 \):
The solutions are:
\[
x_1 = 3.0, \quad x_2 = -5.0
\]

### 3. \( 2x^2 - 3x - 2 = 0 \):
The solutions are:
\[
x_1 = 2.0, \quad x_2 = -0.5
\]

### 4. \( 36x^2 - 60x + 25 = 0 \):
This equation has one real solution (a repeated root):
\[
x_1 = x_2 = 0.83
\]

Would you like more details or have any specific questions on these solutions?

Here are 5 related questions:
1. How do you interpret complex solutions in real-world applications?
2. Can you factor quadratic equations instead of using the quadratic formula?
3. What is the significance of the discriminant in quadratic equations?
4. How do repeated roots affect the shape of a parabola?
5. What are some methods to approximate solutions for complex equations?

**Tip**: Always check the discriminant first to know if you'll have real or complex solutions.

Solve the following quadratic equations: x^2 - 5x + 9 = 0, 4x^2 + 8x - 60 = 0, 2x^2 - 2 - 3x = 0, 36x^2 + 25 = 60x

Learn how to solve various quadratic equations, including those with real and complex roots. This detailed step-by-step solution is perfect for students in Grades 9-10, covering key concepts like the quadratic formula and factoring techniques.

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