SOLUTION: Solve. (1)x squared - 7x - 1 = 0 (2)x squared - 4x - 4 = 0 (3)6x squared = 9x

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Question 122759: Solve.
(1)x squared - 7x - 1 = 0

(2)x squared - 4x - 4 = 0

(3)6x squared = 9x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

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%5E2-7%2Ax-1=0 ( notice a=1, b=-7, and c=-1)




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



x+=+%287+%2B-+sqrt%28+%28-7%29%5E2-4%2A1%2A-1+%29%29%2F%282%2A1%29 Negate -7 to get 7



x+=+%287+%2B-+sqrt%28+49-4%2A1%2A-1+%29%29%2F%282%2A1%29 Square -7 to get 49 (note: remember when you square -7, you must square the negative as well. This is because %28-7%29%5E2=-7%2A-7=49.)



x+=+%287+%2B-+sqrt%28+49%2B4+%29%29%2F%282%2A1%29 Multiply -4%2A-1%2A1 to get 4



x+=+%287+%2B-+sqrt%28+53+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%287+%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+=+%287+%2B-+sqrt%2853%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%287+%2B+sqrt%2853%29%29%2F2 or x+=+%287+-+sqrt%2853%29%29%2F2


Now break up the fraction


x=%2B7%2F2%2Bsqrt%2853%29%2F2 or x=%2B7%2F2-sqrt%2853%29%2F2


Simplify


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


So these expressions approximate to

x=7.14005494464026 or x=-0.140054944640259


So our solutions are:
x=7.14005494464026 or x=-0.140054944640259

Notice when we graph x%5E2-7%2Ax-1, we get:



when we use the root finder feature on a calculator, we find that x=7.14005494464026 and x=-0.140054944640259.So this verifies our answer








# 2


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%5E2-4%2Ax-4=0 ( notice a=1, b=-4, and c=-4)




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



x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%2A1%2A-4+%29%29%2F%282%2A1%29 Negate -4 to get 4



x+=+%284+%2B-+sqrt%28+16-4%2A1%2A-4+%29%29%2F%282%2A1%29 Square -4 to get 16 (note: remember when you square -4, you must square the negative as well. This is because %28-4%29%5E2=-4%2A-4=16.)



x+=+%284+%2B-+sqrt%28+16%2B16+%29%29%2F%282%2A1%29 Multiply -4%2A-4%2A1 to get 16



x+=+%284+%2B-+sqrt%28+32+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%284+%2B-+4%2Asqrt%282%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+=+%284+%2B-+4%2Asqrt%282%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%284+%2B+4%2Asqrt%282%29%29%2F2 or x+=+%284+-+4%2Asqrt%282%29%29%2F2


Now break up the fraction


x=%2B4%2F2%2B4%2Asqrt%282%29%2F2 or x=%2B4%2F2-4%2Asqrt%282%29%2F2


Simplify


x=2%2B2%2Asqrt%282%29 or x=2-2%2Asqrt%282%29


So these expressions approximate to

x=4.82842712474619 or x=-0.82842712474619


So our solutions are:
x=4.82842712474619 or x=-0.82842712474619

Notice when we graph x%5E2-4%2Ax-4, we get:



when we use the root finder feature on a calculator, we find that x=4.82842712474619 and x=-0.82842712474619.So this verifies our answer








# 3


6x%5E2=9x Start with the given equation



6x%5E2-9x=0 Subtract 9x from both sides



3x%282x-3%29=0 Factor the left side



Now set each factor equal to zero:
3x=0 or 2x-3=0

x=0 or x=3%2F2 Now solve for x in each case


So our answer is
x=0 or x=3%2F2


Notice if we graph y=6x%5E2-9x we can see that the roots are x=0 and x=3%2F2 . So this visually verifies our answer.


+graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C0%2C+6x%5E2-9x%29+