# Thread: Form (x + a)^2 +b

1. ## Discriminant

What does the term 'Discriminant' mean?

The question is: Calculate the Discriminant of $\displaystyle 3x^2 + 5x +8$

2. The discriminant of a polynomial of the form $\displaystyle ax^2 + bx + c$, is $\displaystyle b^2 -4ac$. The discriminant is an important feature which, among other things, helps us to determine the nature of the roots of the polynomial.

In your case, you have $\displaystyle 3x^2 + 5x + 8$, so the discriminant will be $\displaystyle 5^2- 4 \cdot 3 \cdot 8 = 25 - 96 = -71$.

3. Recall that the solutions to the quadratic equation $\displaystyle ax^2+ bx+ c= 0$ are, by the quadratic formula, $\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$. Whether there are two real roots, one double root, or two complex roots depends upon whether the number in the square root, $\displaystyle b^2- 4ac$ is positive, zero, or negative.

That number, $\displaystyle b^2- 4ac$ is the "discriminant".

As row said, in your case, the discriminant is -71. That tells us that the equation $\displaystyle 3x^2+ 5x+ 8= 0$ has two complex roots.

It als tells us that the graph of $\displaystyle y= 3x^2+ 5x+ 8$ does not cross the x-axis but is entirely in the upper half-plane.

4. Originally Posted by HallsofIvy
Recall that the solutions to the quadratic equation $\displaystyle ax^2+ bx+ c= 0$ are, by the quadratic formula, $\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$. Whether there are two real roots, one double root, or two complex roots depends upon whether the number in the square root, $\displaystyle b^2- 4ac$ is positive, zero, or negative.

That number, $\displaystyle b^2- 4ac$ is the "discriminant".

As row said, in your case, the discriminant is -71. That tells us that the equation $\displaystyle 3x^2+ 5x+ 8= 0$ has two complex roots.

It als tells us that the graph of $\displaystyle y= 3x^2+ 5x+ 8$ does not cross the x-axis but is entirely in the upper half-plane.
So, how do you find the intersection of the curve?

5. Originally Posted by Mist
So, how do you find the intersection of the curve?
Intersection of the curve with what (note that you've been told in post #3 that it does NOT intersect the x-axis ....)