Question

prove that the roots of the equation x^2-2ax+a^2-b^2-c^2=0 are always real, a,b,cis real number ​

Answer 1

Answer:

Please find the answer in the explanation below=

Step-by-step explanation:

Given equation: $x^{2} - 2ax + a^{2} -b^{2} -c^{2} = 0$

For any quadratic polynomial , the roots are always real if and only if-

D≥ 0

Now, comparing the above equation with $ax^{2} +bx + c =0$ we get-

a=1

b= -2a

c= ($a^{2} -b^{2} - c^2$)

Now D= $b^2- 4ac$

So D= $4a^2$- 4($a^{2} -b^{2} - c^2$)

or D= 4($b^{2}+ c^2$)

Since ($b^{2}+ c^2$) is always ≥ 0

∴ D≥ 0

Hence, the roots of the given equation are always real. Proved!

Hope this is helpful. Please rate the answer!