Maths-

General

Easy

### Question

#### If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:

- both roots in [a, b]
- both roots in
- both roots in
- one root in

### Hint:

The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. It also goes by the name quadratic equations. Here we have to find the roots of equation, (x - a) (x - b) - 1 if b > a.

#### The correct answer is: one root in

#### A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form:

ax² + bx + c = 0

where a, b, and c are constant terms and x is the unknown variable.

Now we have given the equation as: (x - a) (x - b) - 1 = 0. Simplifying it, we get:

x^{2}−(a+b)x−ab−1=0

Now lets find the discriminant. The formula is:

D=b^{2}-4ac

$Applying it, we get:$

$D=(a+b)−4(ab−1)$

$D=(a−b)+4$

$D>0$

$So now we can say that$it has two real roots. Also $f(a)=−1$ and $f(b)=−1$ but $b>a$ which eventually means that $a$ and $b$ are distinct as coefficient of $x$ is position (it is $1$), minima of $f(x)$ is between $a$ and $b.$

Therefore, one of the roots will be in the interval of $(−α,a)$ and the other root will be in the interval $(b,α)$.

Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of $(−α,a)$ and the other root will be in the interval $(b,α)$.

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