Solve by factoring: 6x^2 + x = 5
To solve the equation by factoring, let's first rearrange it so that one side is equal to zero.
6x^2 + x - 5 = 0
To factor the quadratic expression, we need to find two numbers whose product is the product of the coefficient of x^2 (6) and the constant term (-5), and whose sum is equal to the coefficient of x (1).
The factors of the product of 6 and -5 are: 1 and -30, 2 and -15, 3 and -10, 5 and -6.
Since the sum of these factors is not equal to 1, we cannot directly factor the quadratic expression. However, we can use the quadratic formula as an alternative method to solve it.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 6, b = 1, and c = -5. Substituting these values into the quadratic formula, we get:
x = (-1 ± √(1^2 - 4*6*(-5))) / (2*6)
Simplifying further:
x = (-1 ± √(1 + 120)) / 12
x = (-1 ± √121) / 12
Now, let's evaluate the square root:
x = (-1 ± 11) / 12
This gives us two possible solutions:
x = (-1 + 11) / 12 = 10/12 = 5/6
x = (-1 - 11) / 12 = -12/12 = -1
Therefore, the solutions to the equation 6x^2 + x = 5 are x = 5/6 and x = -1.