f(x) = 2x^2 + 5x - 12
First, multiply the first and last number:
2 • -12 = -24
Next, determine which two numbers MULTIPLY to -24 but ADD to the middle number (5)
We find this is -3 and 8 because:
-3 • 8 = -24
And
-3 + 8 = 5
Now we substitute these two factors (-3 and 8) in for the middle term 5:
2x^2 - 3x + 8x - 12
Separate the function into two parts:
2x^2 - 3x
And
8x - 12
Determine the largest common factor of each part and divide it out, then combine them back into one equation:
x(2x - 3) + 4(2x - 3)
The pieces inside the parentheses should always match once you’ve don’t the previous step, this way you know you’ve done it correctly.
Use the part inside the parentheses to solve for one of the “x’s” and the part outside the parentheses to solve for the other “x”:
x + 4
And
2x - 3
Now isolate for each x by setting each equation equal to 0 and isolating x:
x = -4
And
x = 3/2
So x1 = -4 and x2 = 3/2