If the discriminant within the quadratic formula has a zero value. how is this shown when you graph?

This is probably going to be too complicated for the original poster, but I guess this can provide some insight.

Let $m$ and $n$ be two integers greater than $1$, $k$ be a field and let $P\in k[X]_{\leqslant n-1}$ and $Q\in k[X]_{\leqslant m-1}$. Finally, let us consider the following linear transformation: $$\varphi\colon\left\{\begin{array}{ccc}k[X]_{\leqslant m-1}\times k[X]_{\leqslant n-1} & \rightarrow & k[X]_{\leqslant m+n-1}\\(U,V) & \mapsto & PU+QV\end{array}\right..$$ Notice that $\varphi$ is invertible if and only if $P$ and $Q$ are coprime polynomials. Indeed, since $\varphi$ is linear and its domain and codomain have the same dimension over $k$, it suffices to examine when the map $\varphi$ is surjective. The result then follows from Bézout's theorem.

However, $P$ and $Q$ are coprime if and only if they have no common roots in an extension field of $k$.

Finally, $P$ and $Q$ have no common root in an extension field of $k$ if and only if $\varphi$ is invertible if and only if its determinant is nonzero.

Specifying to our case, let $m=3$, $n=2$, $P=aX^2+bX+c$ and $Q=P'=2aX+b$. In that case, let us compute $\det(\varphi)$ in the basis $\{(X^2,0),(X,0),(1,0),(0,X),(0,1)\}$ and $\{X^4,X^3,X^2,X,1\}$, one has: $$\det(\varphi)=\left|\begin{pmatrix}a&0&0&0&0\\b&a&0&0&0\\c&b&a&2a&0\\0&c&b&b&2a\\0&0&c&0&b\end{pmatrix}\right|=a^3(4ac-b^2).$$ Finally, if $a\neq 0$, $P$ is coprime with $P'$ i.e. has simple roots if and only if $b^2-4ac\neq 0$.

If you want to learn more check out resultant and discriminant. Notice that this approach will work for any degree at the price of harder computations.

Students will be able to find the discriminant using the discriminant formula.
Students will be able to determine how many solutions an equation has.

The discriminant is a number that can be calculated from any quadratic equation. When the quadratic equation is in standard form, where a ≠ 0:

Do you remember the reason why the "a" value cannot be equal to zero?

Yes, you are right. If a = 0, then 0 times x^2 would be 0, and the function would be: bx+c=0.

Is the function bx+c=0 quadratic?

No. bx + c = 0 is not a quadratic function. A quadratic function has to be a second degree polynomial, meaning it has an x^2 term.

The formula for the discriminant is:

Do you remember that the solution(s) of a quadratic equation is/are located where the graph intersects the x-axis. These points are also known as zeroes, roots, solutions, and x-intercepts. The discriminant provides critical information regarding the number of the solutions of any quadratic equation prior to solving to find the solutions.

Number of Solutions of a Quadratic Equation:

b² − 4ac > 0, Discriminant is greater than zero, Positive Discriminant:Two Real Solutions

b² − 4ac = 0, Discriminant is equal to zero. One Real Solution

b² − 4ac < 0, Discriminant is less than zero ,Negative Discriminant: No Real Solutions

Notice how the discriminant and number of solutions affects the graph of the quadratic function on the right.

The reason that the number of solutions depends on discriminant will be more clear in next lesson, Quadratic Formula. In short, the discriminant is a part of the Quadratic Formula. Mathematically, using a square root as a part of the formula results in a different number of solutions depending on the sign of the radicand (number under the square root).

Square root of a positive number results in two solutions (+/-), resulting in two solutions. (i.e. sqrt(25) = +5 and -5)
Square root of zero is zero, resulting in only one solution.
Square root of a negative number is undefined as a real number, resulting in no real solutions. (i.e. sqrt(-25) is undefined)

Even without calculating what the roots are, we can find out the number of real solutions just by examining the discriminant of the quadratic function. Let's find the number of real solutions of the following function using discriminant:

First, make sure that the quadratic is in standard form.

This function is in standard form since all terms are on one side of the equation, and the equation is equal to zero.

Next, identify the a, b, and c values.

a=3 b=4 c=-5

Then, substitute into the discriminant formula:

4^2-4 (3) (-5)Finally, simplify using the correct order of operations, PEMDAS.16+60=76 The discriminant is 76, which is positive. This means that there are two real solutions. Watch the screencast on the right for the same example.

Example 1:
9x^2 + 6x = -2

The equation is not in standard form, so we rearrange it.
9x^2 + 6x + 1= -1 + 1 9x^2 + 6x + 1 = 0 We next find the a, b, and c values.

a=9 b=6 c=1

Then, substitute into the discriminant formula.

6^2 - 4(9)(1)

Finally, simplify.
36- 36=0The discriminant is zero, meaning there is one real solution for this quadratic function..

We can check the answer by graphing using a calculator or GeoGebra (see graph on the right).

As you see, there is only one x-intercept, or one real solution.

Example 2:

x^2=x-1

The equation is not in standard form, so we rearrange it.
x^2 - x + 1 = x - x -1 + 1 x^2 - x + 1 = 0

We next find the a, b, and c values.

a=1 b=-1 c=1Then, substitute into the discriminant formula. (-1)^2 - 4(1)(1)

Finally, simplify.

1 - 4 = -3The discriminant is negative, meaning there are no real solutions.

We can check the answer by graphing using a calculator or GeoGebra (see graph on right).

Did you get the same function as below? Does the function touch the x-axis? Is there any solution set for this equation?

Try out the following Quizlet to practice your skills.

Now that you have practiced, try taking the Quizlet Test to ensure your mastery.

Once you have achieved 80% on the Quizlet Test linked above, please progress on to the next lesson, Quadratic Formula.

Do you want some more practice finding the discriminant? Check out the resources below:

If you have any questions regarding this Web-Based Instruction or how to use it, please refer back to the main page or contact your instructor. If you have any questions regarding course content, please contact your instructor using the "Contact" link to the right.

Header photo used under Creative Commons from: Pallavpareek
Khanacademy. "Discriminant of Quadratic Equations." Online video clip. Youtube. Youtube, 12 Mar 2010. Web. 17 June 2013.
Mahalodotcom. "The Quadratic Formula: How to use Discriminant to Determine Roots." Online video clip. Youtube. Youtube, 3 Feb. 2011. Web. 17 June 2013.