SOLUTION: Good Morning Tutor, Could you tell me if I have these problems correct? Solve: x^2 + 7x - 1 = 0 is (-6 +- sqrt of 50 / 2) Solve by completeing the square: 2x

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Good Morning Tutor, Could you tell me if I have these problems correct? Solve: x^2 + 7x - 1 = 0 is (-6 +- sqrt of 50 / 2) Solve by completeing the square: 2x      Log On


   

Question 105112: Good Morning Tutor,
Could you tell me if I have these problems correct?
Solve:
x^2 + 7x - 1 = 0 is (-6 +- sqrt of 50 / 2)
Solve by completeing the square:
2x^2 + 8x - 5 = 0 is (4 +- sqrt of 26 / 2)
Thank you for your help this morning.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

this is how you need to do it:
1.
Solved by pluggable solver: Quadratic Formula
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%2B7%2Ax-1=0 ( notice a=1, b=7, and c=-1)





x+=+%28-7+%2B-+sqrt%28+%287%29%5E2-4%2A1%2A-1+%29%29%2F%282%2A1%29 Plug in a=1, b=7, and c=-1




x+=+%28-7+%2B-+sqrt%28+49-4%2A1%2A-1+%29%29%2F%282%2A1%29 Square 7 to get 49




x+=+%28-7+%2B-+sqrt%28+49%2B4+%29%29%2F%282%2A1%29 Multiply -4%2A-1%2A1 to get 4




x+=+%28-7+%2B-+sqrt%28+53+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-7+%2B-+sqrt%2853%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-7+%2B-+sqrt%2853%29%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%28-7+%2B+sqrt%2853%29%29%2F2 or x+=+%28-7+-+sqrt%2853%29%29%2F2



Now break up the fraction



x=-7%2F2%2Bsqrt%2853%29%2F2 or x=-7%2F2-sqrt%2853%29%2F2



Simplify



x=-7%2F2%2Bsqrt%2853%29%2F2 or x=-7%2F2-sqrt%2853%29%2F2



So the solutions are:

x=-7%2F2%2Bsqrt%2853%29%2F2 or x=-7%2F2-sqrt%2853%29%2F2





2.
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=2+x%5E2%2B8+x-5 Start with the given equation



y%2B5=2+x%5E2%2B8+x Add 5 to both sides



y%2B5=2%28x%5E2%2B4x%29 Factor out the leading coefficient 2



Take half of the x coefficient 4 to get 2 (ie %281%2F2%29%284%29=2).


Now square 2 to get 4 (ie %282%29%5E2=%282%29%282%29=4)





y%2B5=2%28x%5E2%2B4x%2B4-4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 4 does not change the equation




y%2B5=2%28%28x%2B2%29%5E2-4%29 Now factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2



y%2B5=2%28x%2B2%29%5E2-2%284%29 Distribute



y%2B5=2%28x%2B2%29%5E2-8 Multiply



y=2%28x%2B2%29%5E2-8-5 Now add %2B5 to both sides to isolate y



y=2%28x%2B2%29%5E2-13 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=-2, and k=-13. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=2x%5E2%2B8x-5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2%2B8x-5%29 Graph of y=2x%5E2%2B8x-5. Notice how the vertex is (-2,-13).



Notice if we graph the final equation y=2%28x%2B2%29%5E2-13 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x%2B2%29%5E2-13%29 Graph of y=2%28x%2B2%29%5E2-13. Notice how the vertex is also (-2,-13).



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