(D) Let $f(x) = (x - a)(x - b) - 1$.Since $f(x)$ is a downward-opening parabola if we consider the sign,but here the coefficient of $x^2$ is $1$,so it

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One root in $(-\infty, a)$ and the other in $(b, +\infty)$

(D) Let $f(x) = (x - a)(x - b) - 1$.Since $f(x)$ is a downward-opening parabola if we consider the sign,but here the coefficient of $x^2$ is $1$,so it is an upward-opening parabola.Calculate the values at $x = a$ and $x = b$:$f(a) = (a - a)(a - b) - 1 = -1$$f(b) = (b - a)(b - b) - 1 = -1$Since $f(a) Therefore,one root lies in $(-\infty, a)$ and the other root lies in $(b, +\infty)$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

Medium

If $b > a$,then in which interval do the roots of the equation $(x - a)(x - b) - 1 = 0$ lie?

A
Both roots in $[a, b]$
B
Both roots in $(-\infty, a)$
C
Both roots in $(b, +\infty)$
D
One root in $(-\infty, a)$ and the other in $(b, +\infty)$

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

If $f:[1, 2] \rightarrow R$ defined by $f(x) = x^2 + 2kx + k$ is always negative for all $x \in [1, 2]$,then the interval in which $k$ lies is:

MediumAP EAMCET 2017

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For what interval of $m$ do all roots of the quadratic equation $x^2 - 2mx + m^2 - 1 = 0$ lie between $-2$ and $4$?

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Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

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The value of $p$ such that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and the other lies between $2$ and $3$,lies in the interval:

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The number of integral values of $k$,for which one root of the equation $2x^2-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$,is:

DifficultJEE MAIN 2023

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If $b > a$,then in which interval do the roots of the equation $(x - a)(x - b) - 1 = 0$ lie?

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If $f:[1, 2] \rightarrow R$ defined by $f(x) = x^2 + 2kx + k$ is always negative for all $x \in [1, 2]$,then the interval in which $k$ lies is:

For what interval of $m$ do all roots of the quadratic equation $x^2 - 2mx + m^2 - 1 = 0$ lie between $-2$ and $4$?

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

The value of $p$ such that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and the other lies between $2$ and $3$,lies in the interval:

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