The discriminant is part of the quadratic formula. In the quadratic formula, 

$\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The discriminant is the term:

$\displaystyle b^2-4ac$

If the discriminant is 0, there is only one real solution. This would be:

$\displaystyle \frac{-b}{2a}$, since the our discriminant is gone.

If the discriminant is a positive number, then we have two real roots, the usual form of the quadratic equation:

$\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Finally, if the discriminant is negative, we would be taking the square root of a negative number. This will give us no real zeros.

Plugging the numbers into the discriminant gives us:

$\displaystyle 8^2-4(1)(16)$

$\displaystyle 64-64=0$

The discriminant is zero, so there is only one root,

$\displaystyle \frac{-b}{2a}=\frac{-8}{2}=-4$

Using the discriminant, which for a polynomial 

$\displaystyle ax^2+bx+c$

is equal to

$\displaystyle b^2-4ac$,

we can determine the number of roots a polynomial has. If the discriminant is positive, then the polynomial has two real roots. If it is equal to zero, the polynomial has one real root. If it is negative, then the polynomial has two roots which are complex conjugates of one another.

For our function, we have

$\displaystyle a=1, b=7, c=9$,

so when we plug these into the discriminant formula, we get

$\displaystyle 49-4(9)(1)=13>0$

So, our polynomial has two real roots.

The discriminant is found using the following formula:

$\displaystyle b^2-4ac$

For the particular function in question the variable are as follows.

$\displaystyle \\a=1 \\b=-3 \\c=1$

Therefore:

$\displaystyle (-3)^2-4(1)(1)=5$

To determine the amount of roots a given quadratic function has, we must find the discriminant, which for 

$\displaystyle f(x)=ax^2+bx+c$

is equal to

$\displaystyle d=b^2-4ac$

If d is negative, then we have two roots that are complex conjugates of one another. If d is positive, than we have two real roots, and if d is equal to zero, then we have only one real root.

Using our function and the formula above, we get

$\displaystyle d=(6)^2-4(1)(9)=0$

Thus, the function has only one real root.

Write the formula for the discriminant.  This is the term inside the square root of the quadratic formula.

$\displaystyle D=b^2-4ac$

The given equation is already in the form of $\displaystyle y=ax^2+bx+c$.

Substitute the terms into the formula.

$\displaystyle D=(4)^2-4(1)(14) = 16-56 =-40$

The answer is:  $\displaystyle -40$

The polynomial is written in the form $\displaystyle y=ax^2+bx+c$, where

$\displaystyle a=1, b=0, c=6$

Write the formula for the discriminant.  This is the term inside the square root value of the quadratic equation.

$\displaystyle D=b^2-4ac$

Substitute all the knowns into this equation.

$\displaystyle D=b^2-4ac = 0-4(1)(6) = -24$

The answer is:  $\displaystyle -24$

We use the quadratic formula to evaluate the types of roots, but it is not necessary to solve the whole equation. Simply look at the discriminant or square root part. 

$\displaystyle x^2+3x+8=0$

$\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Plug in the correct numbers in the discriminant and simplify. Do not take the square root.

$\displaystyle x=\frac{-b\pm \sqrt{3^2-4(1)8}}{2a}$

Since $\displaystyle 3^2-32$ will be negative, this equation will have two complex solutions.

Discrimant<0= Two complex solutions

Discriminant>0= Two distinct and real solutions

Discriminant=0 = One repeated solution

This equation is already in the form of $\displaystyle y=ax^2+bx+c$.

$\displaystyle a=-\frac{1}{2}$

$\displaystyle b=2$

$\displaystyle c=6$

Write the expression for the discriminant.  This is the term inside the square root of the quadratic equation.

$\displaystyle b^2-4ac$

Substitute the terms into the expression and solve.

$\displaystyle (2)^2-4(-\frac{1}{2})(6) = 4+12 = 16$

The answer is:  $\displaystyle 16$

Identify the coefficients for the polynomial $\displaystyle y=ax^2+bx+c$.

$\displaystyle a=-3$

$\displaystyle b=0$

$\displaystyle c=2$

Write the expression for the discriminant.  This is the expression inside the square root from the quadratic formula.

$\displaystyle D=b^2-4ac$

Substitute the numbers.

$\displaystyle D=(0)^2-4(-3)(2) = 24$

The answer is:  $\displaystyle 24$

The discriminant is the term inside the square root of the quadratic equation.

$\displaystyle b^2-4ac$

Identify the values of each variable.  This corresponds to the coefficients of the quadratic in standard form:

$\displaystyle y=ax^2+bx+c$

$\displaystyle a=1, b=-1,c=-1$

Substitute these values into the expression to determine the discriminant.

$\displaystyle b^2-4ac = (-1)^2-4(1)(-1) = 1+4 = 5$

The discriminant is:  $\displaystyle 5$