Solve each of the following equatins :
`6x^(2)+40=31x`

Answer

To solve the quadratic equation \(6x^2 + 40 = 31x\), we will follow these steps:

### Step 1: Rearrange the equation
We start by moving all terms to one side of the equation to set it to zero.

\[ 6x^2 - 31x + 40 = 0 \]

### Step 2: Identify coefficients
In the equation \(6x^2 - 31x + 40 = 0\), we identify the coefficients:
- \(a = 6\)
- \(b = -31\)
- \(c = 40\)

### Step 3: Use the middle term splitting method
We need to split the middle term \(-31x\) into two terms that add up to \(-31x\) and multiply to \(6 \times 40 = 240\).

We can split \(-31x\) into \(-15x\) and \(-16x\) because:
\[ -15 - 16 = -31 \quad \text{and} \quad -15 \times -16 = 240 \]

### Step 4: Rewrite the equation
Now we can rewrite the equation using the split terms:

\[ 6x^2 - 15x - 16x + 40 = 0 \]

### Step 5: Factor by grouping
Next, we group the terms:

\[ (6x^2 - 15x) + (-16x + 40) = 0 \]

Now, we factor out the common factors from each group:

\[ 3x(2x - 5) - 8(2x - 5) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((2x - 5)\):

\[ (3x - 8)(2x - 5) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor equal to zero:

1. \(3x - 8 = 0\)
2. \(2x - 5 = 0\)

### Step 8: Solve for \(x\)
Now we solve each equation for \(x\):

1. From \(3x - 8 = 0\):
\[ 3x = 8 \implies x = \frac{8}{3} \]

2. From \(2x - 5 = 0\):
\[ 2x = 5 \implies x = \frac{5}{2} \]

### Final Solutions
The solutions to the quadratic equation \(6x^2 + 40 = 31x\) are:
\[ x = \frac{8}{3} \quad \text{and} \quad x = \frac{5}{2} \]

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