Solve: `4sqrt(3)x^(2)+5x-2sqrt(3)=0.`

Options

A. `x= (2sqrt(3))/(3)" or "x=(-sqrt(3))/(4).`
B. `x= (-2sqrt(3))/(3)" or "x=(-sqrt(3))/(4).`
C. `x= (2sqrt(3))/(3)" or "x=(sqrt(3))/(4).`
D. `x= (-2sqrt(3))/(3)" or "x=(sqrt(3))/(4).`

Explanation

To solve the quadratic equation \( 4\sqrt{3}x^{2} + 5x - 2\sqrt{3} = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 4\sqrt{3} \)
- \( b = 5 \)
- \( c = -2\sqrt{3} \)

### Step 2: Calculate the product \( ac \)
Next, we need to calculate the product \( ac \):
\[ ac = (4\sqrt{3})(-2\sqrt{3}) = -8 \cdot 3 = -24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( -24 \) (the value of \( ac \)) and add up to \( 5 \) (the value of \( b \)). The numbers are \( 8 \) and \( -3 \) because:
\[ 8 \cdot (-3) = -24 \quad \text{and} \quad 8 + (-3) = 5 \]

### Step 4: Rewrite the middle term
We can rewrite the equation by splitting the middle term using the numbers found:
\[ 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} = 0 \]

### Step 5: Factor by grouping
Now, we will group the terms:
\[ (4\sqrt{3}x^2 + 8x) + (-3x - 2\sqrt{3}) = 0 \]
Factoring each group:
\[ 4x(\sqrt{3}x + 2) - \sqrt{3}(3x + 2) = 0 \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial:
\[ (\sqrt{3}x + 2)(4x - \sqrt{3}) = 0 \]

### Step 7: Set each factor to zero
Now, we set each factor equal to zero:
1. \( \sqrt{3}x + 2 = 0 \)
2. \( 4x - \sqrt{3} = 0 \)

### Step 8: Solve for \( x \)
For the first equation:
\[ \sqrt{3}x = -2 \implies x = -\frac{2}{\sqrt{3}} \]

For the second equation:
\[ 4x = \sqrt{3} \implies x = \frac{\sqrt{3}}{4} \]

### Final Solutions
The solutions to the equation \( 4\sqrt{3}x^{2} + 5x - 2\sqrt{3} = 0 \) are:
\[ x = -\frac{2}{\sqrt{3}} \quad \text{and} \quad x = \frac{\sqrt{3}}{4} \]

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