tripplek7i

2022-01-30

Find the value of the product of roots of this quadratic equation

It is given that one of the roots of the quadratic equation :

${x}^{2}+(p+3)x-{p}^{2}=0$

It is given that one of the roots of the quadratic equation :

coolbananas03ok

Beginner2022-01-31Added 20 answers

Step 1

Without applying the factorization of a "difference of two squares" or Viete's relations, we can still use the information stated in the problem. If we call the two roots of the quadratic equation r and -r , then we have

and

This means that

So either

But if

which would then make the quadratic equation

But that polynomial factors as

so we couldn't have both roots equal to zero.

Instead, it must be that

for which the roots are given by

the product of the roots is thus -9

Step 2

Another way to arrive at this conclusion is that

is the equation of an "upward-opening" parabola, for which we want the x-intercepts to be

(The vertex is definitely "below" the x-axis at

lirwerwammete9t

Beginner2022-02-01Added 9 answers

Step 1

We know that this equation has at most two roots in the set of reals.

Let's denote them with a and -a.

Then your equation is

${x}^{2}-{a}^{2}=0$ .

Therefore p must be equal to -3.

Hence the equation is

${x}^{2}-9=0$

and the roots are 3 and -3. So the desired product is -9.

We know that this equation has at most two roots in the set of reals.

Let's denote them with a and -a.

Then your equation is

Therefore p must be equal to -3.

Hence the equation is

and the roots are 3 and -3. So the desired product is -9.