Chapter 3.3, Problem 92AYU

To determine

To explain: the possibilities of intersection of two quadratic functions

Answer to Problem 92AYU

Two quadratic functions can intersect at two points or one point or no point.

Explanation of Solution

Two different quadratic functions intersect $1$ time or $2$ times or don’t intersect.

Let us consider the following quadratic functions

  $f(x)=a{x}^2+bx+c$ and

  $g\left(x\right)=d{x}^2+hx+k$

Now, solve them for getting point of intersection as

  $\begin{array}{l}f\left(x\right)=g\left(x\right)\\ a{x}^2+bx+c=d{x}^2+hx+k\\ \left(a-d\right)x^2+\left(b-h\right)x+c-k=0\end{array}$

So, a quadratic equation is obtained.

These cases arise here.

Case - $I$ : When $a\ne d$ , then this will be solved like a quadratic equation, thereby giving us a maximum of two possible roots and hence two possible points of intersection.

Case- $II$ : If $a=d$ , the quadratic equation will become a linear equation.

Now a linear equation has a maximum of one solution, and hence one possible point of intersection.

Case- $III$ : If $a=d$ and $b=h$ , there will not be any quadratic or linear equation, and hence no solution. Therefore, there will not be any point of intersection as well.

Hence, two quadratic functions can intersect at two points or one point or no point.

Conclusion:

Hence, two quadratic functions can intersect at two points or one point or no point.