The expression in the image is:

$\frac{x^2}{4} + x - 4x^2 - \frac{3x}{4} = 13$

To simplify and solve this equation, let's follow these steps:

Step 1: Combine like terms

1. Rewrite the terms involving $x^2$ and $x$ separately.

   • Terms with $x^2$: $\frac{x^2}{4} - 4x^2$
   • Terms with $x$: $x - \frac{3x}{4}$

2. Simplify each part:

   • For the $x^2$ terms: $\frac{x^2}{4} - 4x^2 = \frac{x^2 - 16x^2}{4} = \frac{-15x^2}{4}$

   • For the $x$ terms: $x - \frac{3x}{4} = \frac{4x - 3x}{4} = \frac{x}{4}$

3. Substitute back into the equation: $\frac{-15x^2}{4} + \frac{x}{4} = 13$

Step 2: Clear the fractions by multiplying through by 4

$-15x^2 + x = 52$

Step 3: Rearrange into standard quadratic form

$-15x^2 + x - 52 = 0$

or multiply by $-1$ for simplicity:

$15x^2 - x + 52 = 0$

Step 4: Solve the quadratic equation

The quadratic equation $15x^2 - x + 52 = 0$ can be solved using the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 15 \cdot 52}}{2 \cdot 15}$

Calculate the discriminant: $(-1)^2 - 4 \cdot 15 \cdot 52 = 1 - 3120 = -3119$

Since the discriminant is negative ($-3119$), there are no real solutions for $x$. Therefore, the equation has complex solutions.

Would you like more details on how to handle complex solutions or further explanation of any step?

Related Questions:

1. How do you solve quadratic equations with complex solutions?
2. Can you simplify fractions in equations before solving?
3. How does the quadratic formula apply to equations with no real solutions?
4. Why is the discriminant important in determining solution types?
5. What steps can we take to avoid fractions when solving quadratic equations?

Tip:

When you see fractions in an equation, try clearing them early by multiplying through by the denominator to simplify calculations.