SOLUTION: Use the quadratic formula to solve the equation. -2x^2+x+8=0 Solve the equation. x^2+18x+81=25 Simplify (square root of -175) using the imaginary number i.

Algebra ->  Rational-functions -> SOLUTION: Use the quadratic formula to solve the equation. -2x^2+x+8=0 Solve the equation. x^2+18x+81=25 Simplify (square root of -175) using the imaginary number i.      Log On


   

Question 119412: Use the quadratic formula to solve the equation.
-2x^2+x+8=0
Solve the equation.
x^2+18x+81=25
Simplify (square root of -175) using the imaginary number i.
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 -2%2Ax%5E2%2Bx%2B8=0 ( notice a=-2, b=1, and c=8)




x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A-2%2A8+%29%29%2F%282%2A-2%29 Plug in a=-2, b=1, and c=8



x+=+%28-1+%2B-+sqrt%28+1-4%2A-2%2A8+%29%29%2F%282%2A-2%29 Square 1 to get 1



x+=+%28-1+%2B-+sqrt%28+1%2B64+%29%29%2F%282%2A-2%29 Multiply -4%2A8%2A-2 to get 64



x+=+%28-1+%2B-+sqrt%28+65+%29%29%2F%282%2A-2%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-1+%2B-+sqrt%2865%29%29%2F%282%2A-2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-1+%2B-+sqrt%2865%29%29%2F-4 Multiply 2 and -2 to get -4

So now the expression breaks down into two parts

x+=+%28-1+%2B+sqrt%2865%29%29%2F-4 or x+=+%28-1+-+sqrt%2865%29%29%2F-4


Now break up the fraction


x=-1%2F-4%2Bsqrt%2865%29%2F-4 or x=-1%2F-4-sqrt%2865%29%2F-4


Simplify


x=1+%2F+4-sqrt%2865%29%2F4 or x=1+%2F+4%2Bsqrt%2865%29%2F4


So these expressions approximate to

x=-1.76556443707464 or x=2.26556443707464


So our solutions are:
x=-1.76556443707464 or x=2.26556443707464

Notice when we graph -2%2Ax%5E2%2Bx%2B8, we get:



when we use the root finder feature on a calculator, we find that x=-1.76556443707464 and x=2.26556443707464.So this verifies our answer





#2

x%5E2%2B18x%2B81=25 Start with the given equation


x%5E2%2B18x%2B81-25=0 Subtract 25 from both sides


x%5E2%2B18%2Ax%2B56=0 Combine like 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 x%5E2%2B18%2Ax%2B56=0 ( notice a=1, b=18, and c=56)




x+=+%28-18+%2B-+sqrt%28+%2818%29%5E2-4%2A1%2A56+%29%29%2F%282%2A1%29 Plug in a=1, b=18, and c=56



x+=+%28-18+%2B-+sqrt%28+324-4%2A1%2A56+%29%29%2F%282%2A1%29 Square 18 to get 324



x+=+%28-18+%2B-+sqrt%28+324%2B-224+%29%29%2F%282%2A1%29 Multiply -4%2A56%2A1 to get -224



x+=+%28-18+%2B-+sqrt%28+100+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-18+%2B-+10%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-18+%2B-+10%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%28-18+%2B+10%29%2F2 or x+=+%28-18+-+10%29%2F2

Lets look at the first part:

x=%28-18+%2B+10%29%2F2

x=-8%2F2 Add the terms in the numerator
x=-4 Divide

So one answer is
x=-4



Now lets look at the second part:

x=%28-18+-+10%29%2F2

x=-28%2F2 Subtract the terms in the numerator
x=-14 Divide

So another answer is
x=-14

So our solutions are:
x=-4 or x=-14

Notice when we graph x%5E2%2B18%2Ax%2B56, we get:

+graph%28+500%2C+500%2C+-24%2C+6%2C+-24%2C+6%2C1%2Ax%5E2%2B18%2Ax%2B56%29+

and we can see that the roots are x=-4 and x=-14. This verifies our answer






Solved by pluggable solver: Simplifying Square Roots (whole numbers only)


sqrt%28-175%29 Start with the given expression


sqrt%28-1%2A175%29 Factor out a negative 1


sqrt%28-1%29%2Asqrt%28175%29 Break up the square roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29


i%2Asqrt%28175%29 Replace sqrt%28-1%29 with i (remember i=sqrt%28-1%29)



Now lets simplify sqrt%28175%29:






The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.

So let's list the factors of 175

Factors:

1, 5, 7, 25, 35, 175



Notice how 25 is the largest perfect square, so lets factor 175 into 25*7





sqrt%2825%2A7%29 Factor 175 into 25*7



sqrt%2825%29%2Asqrt%287%29 Break up the square roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29



5%2Asqrt%287%29 Take the square root of the perfect square 25 to get 5



So the expression sqrt%28175%29 simplifies to 5%2Asqrt%287%29




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Answer:

So the expression

sqrt%28-175%29


simplifies to


5%2Ai%2Asqrt%287%29 (just reintroduce i back in)