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# What is the solution set of the following equation? 6x^{2} = 13x + 5

**Solution:**

Given equation is

6x^{2} = 13x + 5.

Subtract 13x from both sides.

6x^{2} - 13x = 5

Subtract 5 from both sides.

6x^{2} - 13x - 5 = 0

To solve the equation, we need to factor.

a + b = -13ab = 6(-5) = -30

Since ab is negative, a and b have the opposite signs. Since a + b is negative, the negative number has greater absolute value than the positive.

List all such integer pairs that give product - 30.

(1, -30), (2, -15), (3, -10), (5, -6)

Calculate the sum for each pair.

1 - 30 = -29

2 - 15 = -13

3 - 10 = -7

5 - 6 = -1

The solution is the pair that gives sum -13

a = -15 b = 2

Rewrite,

(6x^{2} - 15x) + (2x - 5).

Factor out 3x in 6x^{2} - 15x.

3x(2x - 5) + 2x - 5

Factor out common term by using distributive property

(2x - 5) (3x +1)

Solving the above, we get,

x = 5/2 and -1/3.

Therefore, x = 5/2 and -1/3.

## What is the solution set of the following equation? 6x^{2 }= 13x + 5

**Summary:**

The solution set of the following equation 6x^{2} = 13x + 5 is (5/2, -1/3).

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