SOLUTION: Show EACH step in solving each of the following equations. State if any solution is extraneous. a. sqrt(3x + 1) = 4 b. 2x - sqrt(3x - 2) = 8 c. sqrt(2x - 1) = x - 8

Algebra ->  Square-cubic-other-roots -> SOLUTION: Show EACH step in solving each of the following equations. State if any solution is extraneous. a. sqrt(3x + 1) = 4 b. 2x - sqrt(3x - 2) = 8 c. sqrt(2x - 1) = x - 8       Log On


   

Question 39246: Show EACH step in solving each of the following equations. State if any solution is extraneous.
a. sqrt(3x + 1) = 4
b. 2x - sqrt(3x - 2) = 8
c. sqrt(2x - 1) = x - 8

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
a. sqrt(3x + 1) = 4
Square both sides to get:
3x+1=16
3x=15
x=5
b. 2x - sqrt(3x - 2) = 8
Rewrite as sqrt(3x-2)=2x-8
Square both sides to get:
3x-2=4x^2-32x+64
4x^2-35x+66=0
Solve for "x" as follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-35x%2B66+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-35%29%5E2-4%2A4%2A66=169.

Discriminant d=169 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--35%2B-sqrt%28+169+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-35%29%2Bsqrt%28+169+%29%29%2F2%5C4+=+6
x%5B2%5D+=+%28-%28-35%29-sqrt%28+169+%29%29%2F2%5C4+=+2.75

Quadratic expression 4x%5E2%2B-35x%2B66 can be factored:
4x%5E2%2B-35x%2B66+=+4%28x-6%29%2A%28x-2.75%29
Again, the answer is: 6, 2.75. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-35%2Ax%2B66+%29

The solutions x=6 or x=2.75 are both valid.
c. sqrt(2x - 1) = x - 8
Square both sides to get:
2x-1=x^2-16x+64
x^2-18x+65=0
Solve for "x" as follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-18x%2B65+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-18%29%5E2-4%2A1%2A65=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--18%2B-sqrt%28+64+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-18%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+13
x%5B2%5D+=+%28-%28-18%29-sqrt%28+64+%29%29%2F2%5C1+=+5

Quadratic expression 1x%5E2%2B-18x%2B65 can be factored:
1x%5E2%2B-18x%2B65+=+1%28x-13%29%2A%28x-5%29
Again, the answer is: 13, 5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-18%2Ax%2B65+%29

The solutions x=13 or x=5 are both valid.
Cheers,
Stan H.