SOLUTION: How do I solve the equation by completing the sqaure: x^2-4x-10=0 Also, I need to solve -9x-3x^2=5 using the quadratic formula. What's the difference of an algebra problem being

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: How do I solve the equation by completing the sqaure: x^2-4x-10=0 Also, I need to solve -9x-3x^2=5 using the quadratic formula. What's the difference of an algebra problem being      Log On


   

Question 91118: How do I solve the equation by completing the sqaure: x^2-4x-10=0
Also, I need to solve -9x-3x^2=5 using the quadratic formula. What's the difference of an algebra problem being solved by using a quadratic function, and one being solved by using the quadratic formula?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"How do I solve the equation by completing the sqaure: x^2-4x-10=0"


x%5E2-4x-10 Start with the given equation


x%5E2-4x=10 Add 10 to both sides

1%28x%5E2-4x%29=10 Factor out 1



Take half of -4 to get -2 (ie -4%2F2=-2)
Now square -2 to get 4 (ie -2%5E2=4)



1%28x%5E2-4x%2B4%29=10 Add this result (4) inside the parenthesis

1%28x%5E2-4x%2B4%29=10%2B4%281%29 Add 4(1) to the other side (remember we factored out a 1)

Now the left side is a complete square

1%28x-2%29%5E2=10%2B4%281%29 Factor the left side

1%28x-2%29%5E2=14 Multiply and combine like terms on the right side

x-2=0%2B-sqrt%2814%29 Take the square root of both sides

x=2%2B-sqrt%2814%29 Add 2 to both sides

So the expression breaks down to
x=2%2Bsqrt%2814%29 or x=2-sqrt%2814%29


So our answer is approximately
x=5.74165738677394 or x=-1.74165738677394

Here is visual proof

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-4x-10%29+ graph of y=x%5E2-4x-10


When we use the root finder feature on a calculator, we would find that the x-intercepts are x=5.74165738677394 and x=-1.74165738677394, so this verifies our answer.


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"Also, I need to solve -9x-3x^2=5 using the quadratic formula"
-9x-3x%5E2=5

-9x-3x%5E2-5=0 Subtract 5 from both sides

-3x%5E2-9x-5=0 Rearrange the 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 -3%2Ax%5E2-9%2Ax-5=0 ( notice a=-3, b=-9, and c=-5)

x+=+%28--9+%2B-+sqrt%28+%28-9%29%5E2-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Plug in a=-3, b=-9, and c=-5



x+=+%289+%2B-+sqrt%28+%28-9%29%5E2-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Negate -9 to get 9



x+=+%289+%2B-+sqrt%28+81-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Square -9 to get 81 (note: remember when you square -9, you must square the negative as well. This is because %28-9%29%5E2=-9%2A-9=81.)



x+=+%289+%2B-+sqrt%28+81%2B-60+%29%29%2F%282%2A-3%29 Multiply -4%2A-5%2A-3 to get -60



x+=+%289+%2B-+sqrt%28+21+%29%29%2F%282%2A-3%29 Combine like terms in the radicand (everything under the square root)



x+=+%289+%2B-+sqrt%2821%29%29%2F%282%2A-3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%289+%2B-+sqrt%2821%29%29%2F-6 Multiply 2 and -3 to get -6

So now the expression breaks down into two parts

x+=+%289+%2B+sqrt%2821%29%29%2F-6 or x+=+%289+-+sqrt%2821%29%29%2F-6


Now break up the fraction


x=%2B9%2F-6%2Bsqrt%2821%29%2F-6 or x=%2B9%2F-6-sqrt%2821%29%2F-6


Simplify


x=-3+%2F+2-sqrt%2821%29%2F6 or x=-3+%2F+2%2Bsqrt%2821%29%2F6


So these expressions approximate to

x=-2.26376261582597 or x=-0.736237384174027


So our solutions are:
x=-2.26376261582597 or x=-0.736237384174027

Notice when we graph -3%2Ax%5E2-9%2Ax-5, we get:



when we use the root finder feature on a calculator, we find that x=-2.26376261582597 and x=-0.736237384174027.So this verifies our answer

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"What's the difference of an algebra problem being solved by using a quadratic function, and one being solved by using the quadratic formula?"

I'm not sure I understand what you're asking here