math question?

Question

When using the quadratic formula to solve a quadratic equation ax2 bx c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero or negative. (When the discriminant is negative, then we have the square root of a negative number. This is called an imaginary number sqrt (-1)= i. Explain what the value of the discriminant means to the graph of y = ax2 bx c. Hint: Choose values of a, b, and c to create a particular discriminant. Then, graph the corresponding equation.

Answer 1

let roots are x1,x2
x1 + x2 = -b/a
x1*x2 = c/a

(x1-x2)^2 = (x1+x2)^2 - 4 * x1*x2
= b^2/4a^2 - 4c/a = (b^2 - 4ac)/4a^2

So |x1 - x2| = sqrt (b^2 - 4ac) / 2*a

|x1 - x2| gives distance between the two roots

Now if b^2 - 4ac>0 then there exists some distance between two roots. So there are two distinct roots.

If b^2 - 4ac = 0 then there is no distance between the roots. So they coincide. So the parabola touches the x axis at only one point.

If b^2 - 4ac <0 then the distance between the roots is imaginary. This implies that we cannot plot the points on real plane, since all distances on real plane are real. We get two imaginary roots.

Answer 2

If the discriminant = 0 then x = -b/2a meaning there is a real double root. The vertex of the parabola will lie on the x-axis. The parabola will be concave downward (have a maximum) if a<0 and concave upward (have a minimum) if a>0.

If the discriminant is <0, then there are no real solutions;the roots will be imaginary. The graph of the parabola will lie entirely above or entirely below the x-axis, never crossing the x-axis.

If the discriminant >0, then there will be two real solutions. The parabola will be concave downward (have a maximum) if a<0 and concave upward (have a minimum) if a>0.

Answer 3

When the discriminant is negative tere is no answer to the equasion.