SOLUTION: 6. Solve using the quadratic formula: x^2 – 5x – 1 = -7

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: 6. Solve using the quadratic formula: x^2 – 5x – 1 = -7       Log On


   

Question 132984This question is from textbook intermediate algebra
: 6. Solve using the quadratic formula:
x^2 – 5x – 1 = -7
This question is from textbook intermediate algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2-5x-1=-7 Start with the given equation


x%5E2-5x-1%2B7=0 Add 7 to both sides.


x%5E2-5x%2B6=0 Combine like terms

Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve x%5E2-5%2Ax%2B6=0 ( notice a=1, b=-5, and c=6)




x+=+%28--5+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A6+%29%29%2F%282%2A1%29 Plug in a=1, b=-5, and c=6



x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A6+%29%29%2F%282%2A1%29 Negate -5 to get 5



x+=+%285+%2B-+sqrt%28+25-4%2A1%2A6+%29%29%2F%282%2A1%29 Square -5 to get 25 (note: remember when you square -5, you must square the negative as well. This is because %28-5%29%5E2=-5%2A-5=25.)



x+=+%285+%2B-+sqrt%28+25%2B-24+%29%29%2F%282%2A1%29 Multiply -4%2A6%2A1 to get -24



x+=+%285+%2B-+sqrt%28+1+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%285+%2B-+1%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%285+%2B-+1%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%285+%2B+1%29%2F2 or x+=+%285+-+1%29%2F2

Lets look at the first part:

x=%285+%2B+1%29%2F2

x=6%2F2 Add the terms in the numerator
x=3 Divide

So one answer is
x=3



Now lets look at the second part:

x=%285+-+1%29%2F2

x=4%2F2 Subtract the terms in the numerator
x=2 Divide

So another answer is
x=2

So our solutions are:
x=3 or x=2

Notice when we graph x%5E2-5%2Ax%2B6, we get:

+graph%28+500%2C+500%2C+-8%2C+13%2C+-8%2C+13%2C1%2Ax%5E2%2B-5%2Ax%2B6%29+

and we can see that the roots are x=3 and x=2. This verifies our answer