Find the roots of the equations.
Q. `6x^(2)+7x-20=0`.

Answer

To find the roots of the quadratic equation \(6x^2 + 7x - 20 = 0\), we can use the method of splitting the middle term. Here’s a step-by-step solution:

### Step 1: Write down the equation
The given equation is:
\[ 6x^2 + 7x - 20 = 0 \]

### Step 2: Identify \(a\), \(b\), and \(c\)
In the quadratic equation \(ax^2 + bx + c = 0\), we have:
- \(a = 6\)
- \(b = 7\)
- \(c = -20\)

### Step 3: Multiply \(a\) and \(c\)
Calculate \(ac\):
\[ ac = 6 \times (-20) = -120 \]

### Step 4: Find two numbers that multiply to \(ac\) and add to \(b\)
We need two numbers that multiply to \(-120\) and add to \(7\). The numbers are \(15\) and \(-8\) because:
\[ 15 \times (-8) = -120 \quad \text{and} \quad 15 + (-8) = 7 \]

### Step 5: Rewrite the middle term
Now, rewrite the equation by splitting the middle term:
\[ 6x^2 + 15x - 8x - 20 = 0 \]

### Step 6: Group the terms
Group the terms:
\[ (6x^2 + 15x) + (-8x - 20) = 0 \]

### Step 7: Factor by grouping
Factor out the common terms in each group:
\[ 3x(2x + 5) - 4(2x + 5) = 0 \]

### Step 8: Factor out the common binomial
Now, factor out the common binomial \((2x + 5)\):
\[ (2x + 5)(3x - 4) = 0 \]

### Step 9: Set each factor to zero
Now, set each factor equal to zero:
1. \(2x + 5 = 0\)
2. \(3x - 4 = 0\)

### Step 10: Solve for \(x\)
Solving the first equation:
\[ 2x + 5 = 0 \implies 2x = -5 \implies x = -\frac{5}{2} \]

Solving the second equation:
\[ 3x - 4 = 0 \implies 3x = 4 \implies x = \frac{4}{3} \]

### Step 11: State the roots
The roots of the equation \(6x^2 + 7x - 20 = 0\) are:
\[ x = -\frac{5}{2} \quad \text{and} \quad x = \frac{4}{3} \]

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