Question

Use the discriminant to determine how many real solutions there are to the function y = -5x2 - 10x - 5.

A. There are no real number solutions.

B. All real numbers are solutions.

C. There are two real number solutions.

D. There is only one real number solution.
4. Use the quadratic formula to determine the solutions for the function y = -2x2 + 3x + 2.
A.There are no real number solutions.
B.x = -1/2, x = 2
C.x = -1/2
D.x = 1/2, x = -2

Answer 1

Remember that the discriminant of a quadratic is the value found under the square root in the quadratic equation.
\[\Delta = b^2-4ac\]Discriminant > 0 means that there are two unique real solutions. Additionally, if it is a perfect square, both solutions will be rational.
Discriminant = 0 means that there is one repeated real solution. (Because sqrt0 = 0, it doesn't matter whether you add or subtract; there will only be one value of x.)
Discriminant < 0 means that there are two imaginary/complex solutions, which are not real solutions.

-10^2 - 4(-5)(-5)
100 - 4(25)
100 - 100
0
Since the discriminant is 0, there is only one real solution.

Remember that the quadratic formula is:
\[x=\frac{-b \pm \sqrt{b^2 -4ac}}{2a}\] For -2x^2+3x+2, a=-2, b=3, and c=2.
Plugging in, we see:
-3 pm sqrt[ 3^2 - 4(-2)(2) ]
x = -------------------------------------
2(-2)
-3 pm sqrt[ 9 - (-16) ]
x = -------------------------------------
-4
-3 pm sqrt[ 25 ]
x = --------------------------
-4
-3 + 5 -3 - 5
x = --------------- , --------------
-4 -4

So x=-1/2 and x=2.