SOLUTION: Good evening, I'm having trouble finding the x-intercept of the following equation: {{{2x^2+16x+9}}} I already managed to complete the square to get the vertex form: {{{2(

Algebra ->  Equations -> SOLUTION: Good evening, I'm having trouble finding the x-intercept of the following equation: {{{2x^2+16x+9}}} I already managed to complete the square to get the vertex form: {{{2(      Log On


   

Question 367314: Good evening, I'm having trouble finding the x-intercept of the following equation:
2x%5E2%2B16x%2B9
I already managed to complete the square to get the vertex form:
2%28x%2B4%29%5E2+-+23
The y-intercept is 9, obtained by plugging 0 in for x and solving for y.
The vertex is (4, -23), but I don't understand how to find the x-intercept. I graphed the equation and the x-intercepts are somewhere around (-7.5, 0) and (-0.6, 0) but I have no idea how to calculate these. Please help! Your time is much, MUCH appreciated :)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
There are two ways to do this. Both methods involve setting each expression equal to zero.

Method 1:

2x%5E2%2B16x%2B9=0 Start with the given equation.


Notice that the quadratic 2x%5E2%2B16x%2B9 is in the form of Ax%5E2%2BBx%2BC where A=2, B=16, and C=9


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


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%2816%29+%2B-+sqrt%28+%2816%29%5E2-4%282%29%289%29+%29%29%2F%282%282%29%29 Plug in A=2, B=16, and C=9


x+=+%28-16+%2B-+sqrt%28+256-4%282%29%289%29+%29%29%2F%282%282%29%29 Square 16 to get 256.


x+=+%28-16+%2B-+sqrt%28+256-72+%29%29%2F%282%282%29%29 Multiply 4%282%29%289%29 to get 72


x+=+%28-16+%2B-+sqrt%28+184+%29%29%2F%282%282%29%29 Subtract 72 from 256 to get 184


x+=+%28-16+%2B-+sqrt%28+184+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%28-16+%2B-+2%2Asqrt%2846%29%29%2F%284%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-16%2B2%2Asqrt%2846%29%29%2F%284%29 or x+=+%28-16-2%2Asqrt%2846%29%29%2F%284%29 Break up the expression.


x+=+%28-8%2Bsqrt%2846%29%29%2F%282%29 or x+=+%28-8-sqrt%2846%29%29%2F%282%29 Reduce.


So the solutions are x+=+%28-8%2Bsqrt%2846%29%29%2F%282%29 or x+=+%28-8-sqrt%2846%29%29%2F%282%29


This means that the x-intercepts are and


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Method 2:


2%28x%2B4%29%5E2-23=0 Start with the given equation.


2%28x%2B4%29%5E2=0%2B23Add 23 to both sides.


2%28x%2B4%29%5E2=23 Combine like terms.


%28x%2B4%29%5E2=%2823%29%2F%282%29 Divide both sides by 2.


x%2B4=%22%22%2B-sqrt%2823%2F2%29 Take the square root of both sides.


x%2B4=sqrt%2823%2F2%29 or x%2B4=-sqrt%2823%2F2%29 Break up the "plus/minus" to form two equations.


x%2B4=sqrt%2846%29%2F2 or x%2B4=-sqrt%2846%29%2F2 Simplify the square root.


x=-4%2Bsqrt%2846%29%2F2 or x=-4-sqrt%2846%29%2F2 Subtract 4 from both sides.


x=%28-8%2Bsqrt%2846%29%29%2F%282%29 or x=%28-8-sqrt%2846%29%29%2F%282%29 Combine the fractions.


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Answer:


So the solutions are x=%28-8%2Bsqrt%2846%29%29%2F%282%29 or x=%28-8-sqrt%2846%29%29%2F%282%29.


So again, the x-intercepts are and


Note: The approximate form of the x-intercepts is x-intercepts are and


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim