#x^2 + 10x = 15#
First, we want to set one side to #0# and let one side have #3# terms so that we can factor it, so we subtract #15# from both sides of the equation:
#x^2 + 10x - 15 = 0#
Now we factor. We have to find two numbers that:
- Multiply up to #-15#
- Add up to #10#.
We know that the factors of #-15# are #-15, -5, -3, -1, 1, 3, 5,# and #15#. However, no group of factors of #-15# can add up to #10#, so we have to do another method, called the quadratic formula.
The quadratic formula is #x = (-b +- sqrt(b^2 - 4ac))/(2a)#.
Our equation is in the form of #ax^2 + bx^2 + c#, which is also called standard form. So we know that:
#a = 1#
#b = 10#
#c = -15#
Now let's substitute these values into the quadratic formula:
#x = (-10 +- sqrt(10^2 - 4(1)(-5)))/(2(1))#
Simplify by doing #10^2#, #-4(1)(-5)#, and #2(1)#:
#x = (-10 +- sqrt(100 + 20))/2#
Add #100 + 20#:
#x = (-10 +- sqrt(120))/2#
Radicalize/simplify #120#
#x = (-10 +- sqrt(4*30))/2#
#x = (-10 +- 2sqrt30)/2#
Divide by #2#:
#x = -5 +- sqrt30#
Hope this helps!
