The expression x^3 - 3x^2 - 9x + c can be wri | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given expression: $x^3 - 3x^2 - 9x + c = (x - a)^2 (x - b)$Expanding the right side: $(x^2 - 2ax + a^2)(x - b) = x^3 - bx^2 - 2ax^2 + 2abx + a^2x

Vedclass · Accepted Answer

$-5$ or $27$

Vedclass · Answer

(A) Given expression: $x^3 - 3x^2 - 9x + c = (x - a)^2 (x - b)$Expanding the right side: $(x^2 - 2ax + a^2)(x - b) = x^3 - bx^2 - 2ax^2 + 2abx + a^2x - a^2b = x^3 - (2a + b)x^2 + (a^2 + 2ab)x - a^2b$Comparing coefficients with $x^3 - 3x^2 - 9x + c$:$2a + b = 3$ $(1)$$a^2 + 2ab = -9$ $(2)$$c = -a^2b$ $(3)$From $(1)$,$b = 3 - 2a$. Substituting into $(2)$:$a^2 + 2a(3 - 2a) = -9$$a^2 + 6a - 4a^2 = -9$$-3a^2 + 6a + 9 = 0$$a^2 - 2a - 3 = 0$$(a - 3)(a + 1) = 0$So,$a = 3$ or $a = -1$.If $a = 3$,$b = 3 - 2(3) = -3$. Then $c = -(3)^2(-3) = 27$.If $a = -1$,$b = 3 - 2(-1) = 5$. Then $c = -(-1)^2(5) = -5$.Thus,$c = -5$ or $27$.

The expression $x^3 - 3x^2 - 9x + c$ can be written in the form $(x - a)^2 (x - b)$ if the value of $c$ is

Similar Questions

If the Arithmetic Mean ($A$.$M$.) and Geometric Mean ($G$.$M$.) of the roots of a quadratic equation in $x$ are $p$ and $q$ respectively,then the equation is:

Find the sum of all possible values of $k$ for which the roots of the equation $x^2 + (k + 1)x + \lambda = 0$ are the square of each other.

If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$,then the quadratic equation whose roots are $\sqrt{5} \alpha, \sqrt{5} \beta$ is

The sum of the roots of a quadratic equation is $2$ and the sum of their cubes is $98$. Then the equation is:

If $2$ and $6$ are the roots of the equation $ax^2 + bx + 1 = 0$,then the quadratic equation,whose roots are $\frac{1}{2a + b}$ and $\frac{1}{6a + b}$,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module