What are the roots of the quadratic equation 15 x ^(2) - 2x - 8 =0

Options

A. `2/3,-4/5`
B. `-2/3,4/5`
C. `-3/2,5/4`
D. `3/2,-5/4`

Answer

To find the roots of the quadratic equation \( 15x^2 - 2x - 8 = 0 \), we can use the method of factorization. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 15 \)
- \( b = -2 \)
- \( c = -8 \)

### Step 2: Calculate the product \( ac \)
We need to calculate \( ac \):
\[ ac = 15 \times (-8) = -120 \]

### Step 3: Find two numbers that add to \( b \) and multiply to \( ac \)
We are looking for two numbers that add up to \( -2 \) (the coefficient \( b \)) and multiply to \( -120 \) (the product \( ac \)).

The two numbers that satisfy these conditions are \( 10 \) and \( -12 \) because:
\[ 10 + (-12) = -2 \quad \text{and} \quad 10 \times (-12) = -120 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the quadratic equation using these two numbers:
\[ 15x^2 + 10x - 12x - 8 = 0 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (15x^2 + 10x) + (-12x - 8) = 0 \]
Now, factor out the common factors from each group:
\[ 5x(3x + 2) - 4(3x + 2) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (3x + 2) \):
\[ (5x - 4)(3x + 2) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor equal to zero:
1. \( 5x - 4 = 0 \)
2. \( 3x + 2 = 0 \)

### Step 8: Solve for \( x \)
For the first equation:
\[ 5x - 4 = 0 \implies 5x = 4 \implies x = \frac{4}{5} \]

For the second equation:
\[ 3x + 2 = 0 \implies 3x = -2 \implies x = -\frac{2}{3} \]

### Conclusion
The roots of the quadratic equation \( 15x^2 - 2x - 8 = 0 \) are:
\[ x = \frac{4}{5} \quad \text{and} \quad x = -\frac{2}{3} \]

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