SOLUTION: Consider the following function. f(x) = 6x2 + 7x − 24 (a) Use the quadratic formula to find the zeros of f. (Enter your answers as a comma-separated list. If an answer does

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Consider the following function. f(x) = 6x2 + 7x − 24 (a) Use the quadratic formula to find the zeros of f. (Enter your answers as a comma-separated list. If an answer does      Log On


   

Question 845169: Consider the following function.
f(x) = 6x2 + 7x − 24
(a) Use the quadratic formula to find the zeros of f. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
x = Correct: Your answer is correct.
(b) Find the maximum or minimum value of f(x). (Round your answer to two decimal places.
Answer by pmesler(52) About Me  (Show Source):
You can put this solution on YOUR website!
By applying the quadratic formula to this equation 6x^2+7x-24 = 0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B7x%2B-24+=+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%2A6%2A-24=625.

Discriminant d=625 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+625+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%287%29%2Bsqrt%28+625+%29%29%2F2%5C6+=+1.5
x%5B2%5D+=+%28-%287%29-sqrt%28+625+%29%29%2F2%5C6+=+-2.66666666666667

Quadratic expression 6x%5E2%2B7x%2B-24 can be factored:
6x%5E2%2B7x%2B-24+=+6%28x-1.5%29%2A%28x--2.66666666666667%29
Again, the answer is: 1.5, -2.66666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B7%2Ax%2B-24+%29


We get (-2.66, 0) and (1.5, 0) as the two roots.
The vertex of of the parabola that is formed from this equation is at the point
(-21, -26.04). Therefore the minimum value for f(x) is -26.04.