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x -intercepts of a Quadratic Function

The x -intercepts of the function f(x)=a x 2 +bx+c,a0 are the solutions of the quadratic equation a x 2 +bx+c=0 . The solutions of a quadratic equation of the form a x 2 +bx+c=0 are given by the quadratic formula.

x= b± b 2 4ac 2a

Here, the expression b 2 4ac is called the discriminant.

The discriminant can be used to confirm the number of x -intercepts and the type of solutions of the quadratic equation.

Value of Discriminant
Type and Number of x -intercepts (roots)
Example of Graph of Related Function

b 2 4ac> 0;

b 2 4ac is a perfect square

2 real and rational

f(x)= x 2 x6

b 2 4ac> 0;

b 2 4ac is not a perfect square

2 real and irrational

f(x)= x 2 3

b 2 4ac=0 1 real

f(x)= (x2) 2

b 2 4ac<0 No x -intercepts

f(x)= x 2 4x+7