SOLUTION: Solve the equation 8x^2+7x=-2 by completing the square

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Question 644009: Solve the equation 8x^2+7x=-2 by completing the square
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

8x%5E2%2B7x=-2+
8x%5E2%2B7x%2B2=0+
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=8+x%5E2%2B7+x%2B2 Start with the given equation



y-2=8+x%5E2%2B7+x Subtract 2 from both sides



y-2=8%28x%5E2%2B%287%2F8%29x%29 Factor out the leading coefficient 8



Take half of the x coefficient 7%2F8 to get 7%2F16 (ie %281%2F2%29%287%2F8%29=7%2F16).


Now square 7%2F16 to get 49%2F256 (ie %287%2F16%29%5E2=%287%2F16%29%287%2F16%29=49%2F256)





y-2=8%28x%5E2%2B%287%2F8%29x%2B49%2F256-49%2F256%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 49%2F256 does not change the equation




y-2=8%28%28x%2B7%2F16%29%5E2-49%2F256%29 Now factor x%5E2%2B%287%2F8%29x%2B49%2F256 to get %28x%2B7%2F16%29%5E2



y-2=8%28x%2B7%2F16%29%5E2-8%2849%2F256%29 Distribute



y-2=8%28x%2B7%2F16%29%5E2-49%2F32 Multiply



y=8%28x%2B7%2F16%29%5E2-49%2F32%2B2 Now add 2 to both sides to isolate y



y=8%28x%2B7%2F16%29%5E2%2B15%2F32 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=8, h=-7%2F16, and k=15%2F32. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=8x%5E2%2B7x%2B2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C8x%5E2%2B7x%2B2%29 Graph of y=8x%5E2%2B7x%2B2. Notice how the vertex is (-7%2F16,15%2F32).



Notice if we graph the final equation y=8%28x%2B7%2F16%29%5E2%2B15%2F32 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C8%28x%2B7%2F16%29%5E2%2B15%2F32%29 Graph of y=8%28x%2B7%2F16%29%5E2%2B15%2F32. Notice how the vertex is also (-7%2F16,15%2F32).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.