SOLUTION: Solve using the quadratic formula: x2 – 3x – 23 = 5

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Question 113753: Solve using the quadratic formula:
x2 – 3x – 23 = 5
Found 2 solutions by checkley71, jim_thompson5910:
Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
X^2-3X-23=5
X^2-3X-23-5=0
X^2-3X-28=0
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
X=(3+-SQRT[(3^2-4*1*-28])/281
X=(3+-SQRT9+112)/2
X=(3+-SQRT121)/2
X=(3+-11)/2
X=(3+11)/2
X=14/2
X=7 ANSWER.
X=(3-11)/2
X=-8/2
X=-4 ANSWER.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2-3%2Ax-23=5 Start with the given equation


x%5E2-3%2Ax-28=0 Subtract 5 from both sides


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%5E2-3%2Ax-28=0 ( notice a=1, b=-3, and c=-28)




x+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A-28+%29%29%2F%282%2A1%29 Plug in a=1, b=-3, and c=-28



x+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A-28+%29%29%2F%282%2A1%29 Negate -3 to get 3



x+=+%283+%2B-+sqrt%28+9-4%2A1%2A-28+%29%29%2F%282%2A1%29 Square -3 to get 9 (note: remember when you square -3, you must square the negative as well. This is because %28-3%29%5E2=-3%2A-3=9.)



x+=+%283+%2B-+sqrt%28+9%2B112+%29%29%2F%282%2A1%29 Multiply -4%2A-28%2A1 to get 112



x+=+%283+%2B-+sqrt%28+121+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%283+%2B-+11%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%283+%2B-+11%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%283+%2B+11%29%2F2 or x+=+%283+-+11%29%2F2

Lets look at the first part:

x=%283+%2B+11%29%2F2

x=14%2F2 Add the terms in the numerator
x=7 Divide

So one answer is
x=7



Now lets look at the second part:

x=%283+-+11%29%2F2

x=-8%2F2 Subtract the terms in the numerator
x=-4 Divide

So another answer is
x=-4

So our solutions are:
x=7 or x=-4

Notice when we graph x%5E2-3%2Ax-28, we get:

+graph%28+500%2C+500%2C+-14%2C+17%2C+-14%2C+17%2C1%2Ax%5E2%2B-3%2Ax%2B-28%29+

and we can see that the roots are x=7 and x=-4. This verifies our answer