Question

The roots of the equation \[(x - a)(x - b) = {b^2}\] are
a.real and equal
b.real and unequal
c.imaginary
d.equal

Answer 1

Hint: We have the given equation where we need to find how the roots are in this equation. So, to start with we will simplify this equation in the simplest way. Then we will try to find the discriminant which will tell us how the roots are.

Complete step-by-step answer:
Given: The equation as: \[(x - a)(x - b) = {b^2}\]
To find: The properties of the roots of the given equation
Now we have given the equation as:
\[(x - a)(x - b) = {b^2}\]
Now expanding it, we get
\[ \Rightarrow {x^2} - ax - bx + ab = {b^2}\]
On rearranging we get,
 \[ \Rightarrow {x^2} - (a + b)x - {b^2} + ab = 0\]
 So, to find the properties of the roots we are going to find the discriminant of the given equation,
Now, the discriminant term which is to be said as, \[D = {b^2} - 4ac\] for a quadratic equation \[a{x^2} + bx + c = 0\]
So, \[D = {( - (a + b))^2} - 4.1.( - {b^2} + ab)\]
On simplification we get,
\[ = {a^2} + 2ab + {b^2} + 4{b^2} - 4ab\]
On adding like terms we get,
\[ = {a^2} - 2ab + {b^2} + 4{b^2}\]
On using, \[{a^2} - 2ab + {b^2} = {(a - b)^2}\], we get
\[ = {(a - b)^2} + {(2b)^2}\]
So, we can see, the discriminant term is the sum of two square terms here, so, both of them are positive. Then we have our discriminant term as positive. Which denotes that the equation will always have real and distinct roots.
So, we have our answer as real and unequal roots which is option b.

Note: In this problem our discriminant, D\[ = {(a - b)^2} + {(2b)^2}\]. We have taken this as greater than or equal to zero, but there can be a case where we may have \[a = b = 0\] which will give us our discriminant is equal to 0. So, we need to keep this case in mind too.