Question

Calculate the discriminant to determine the number of real roots.

y = x^2 + 3x + 9

How many real roots does the equation have?
two real roots
no real roots
one real root
no solution to the equation

Answer 1

when we solve quadratic equations by the quadratic formula, we find the following general form of the solutions to \(ax^2+bx+c=0\): $$x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}$$

Answer 2

the expression under the square root, \(b^2-4ac\), is termed the *discriminant* in that it helps us characterize the types of solutions our quadratic equation has

Answer 3

in particular, when \(b^2-4ac=0\), you'll notice the expression for our solutions reduces to give only a single solution \(x=-\frac{b}{2a}\); this corresponds to the parabola corresponding to our quadratic equation touching the x-axis at only a single point, its vertex. when \(b^2-4ac>0\), we instead get two distinct solutions symmetric about the line of symmetry \(x=-\frac{b}{2a}\), corresponding to the parabola intersecting the x-axis at two distinct points. lastly, when \(b^2-4ac<0\), we find that we have no real solutions (square root of negative numbers is undefined for real numbers) and this corresponds to our parabola not touching the x-axis at all!

Answer 4

in this case, we have \(a=1,b=3,c=9\) so \(b^2-4ac=3^2-4(1)(9)=9-36=-27<0\) and thus we have no real solutions