Find the discriminant of the quadratic equation x2 + 10x + 24 = 0 and use it to determine the number and types of solutions.
b2 - 4ac

2 real solutions since the discriminant is 4 and is greater than 0.

Furthermore, since 4 is a perfect square then the answers are also going to be rational.

You will have 2 real rational solutions.

Step-by-step explanation:

Compare the following to determine the values for :

.

.

Now we are going to compute:

So the discriminant is 4.

Here is the thing about discriminant (D) assuming are real:

If D=0, then you have one real solution.

If D<0, then you have no real solution.

If D>0, then you have two real solutions.

Our D (discriminant) is 4 and 4 is positive or >0, so we have two real solutions.

There is another thing we can tell about this particular discriminant.

Since it is a perfect square, then we know the answers will also be rational.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.