SOLUTION: For each of the following equations use the discriminant to determine what type of number the solutions will be (i.e. rational, irrational, or imaginary), and how many solutions ex

Algebra ->  Radicals -> SOLUTION: For each of the following equations use the discriminant to determine what type of number the solutions will be (i.e. rational, irrational, or imaginary), and how many solutions ex      Log On


   

Question 184704: For each of the following equations use the discriminant to determine what type of number the solutions will be (i.e. rational, irrational, or imaginary), and how many solutions exist. Defend your answers using algebra.

a) 9x^2-12x+4=0
b) 2x2+7x-3=0
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-12x%2B4+=+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=%28-12%29%5E2-4%2A9%2A4=0.

Discriminant d=0 is zero! That means that there is only one solution: x+=+%28-%28-12%29%29%2F2%5C9.
Expression can be factored: 9x%5E2%2B-12x%2B4+=+9%28x-0.666666666666667%29%2A%28x-0.666666666666667%29

Again, the answer is: 0.666666666666667, 0.666666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-12%2Ax%2B4+%29

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B7x%2B-3+=+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=%287%29%5E2-4%2A2%2A-3=73.

Discriminant d=73 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+73+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%287%29%2Bsqrt%28+73+%29%29%2F2%5C2+=+0.386000936329383
x%5B2%5D+=+%28-%287%29-sqrt%28+73+%29%29%2F2%5C2+=+-3.88600093632938

Quadratic expression 2x%5E2%2B7x%2B-3 can be factored:
2x%5E2%2B7x%2B-3+=+2%28x-0.386000936329383%29%2A%28x--3.88600093632938%29
Again, the answer is: 0.386000936329383, -3.88600093632938. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B7%2Ax%2B-3+%29