The two equations x 3-1-0 and ax 2- bx - c -0- a - b - c - R have two roots in common- Then a - b is equal toA- 2B- 0C- -1D- 3

The correct option is B 0x^3+1=0 ⇒ (x+1)(x^2 x+1)=0 x^2 x+1=0 has non real roots as Δ lt;0 We know that, for a quadratic equation with real coefficients, im ...

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Standard XII

Mathematics

Common Roots

The two equat...

Question

The two equations $x^{3} + 1 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R$ have two roots in common. Then $a + b$ is equal to

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Solution

The correct option is B $0$
$x^{3} + 1 = 0 \Rightarrow (x + 1) (x^{2} - x + 1) = 0$
$x^{2} - x + 1 = 0$ has non-real roots as $Δ < 0$

We know that, for a quadratic equation with real coefficients, imaginary roots always occur in conjugate pair.
$∴ x^{2} - x + 1 = 0$ and $a x^{2} + b x + c = 0$ have both roots in common.
$\Rightarrow \frac{1}{a} = \frac{- 1}{b} = \frac{1}{c}$
$\Rightarrow a + b = 0$

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Common Roots

Standard XII Mathematics

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The two equations x3+1=0 and ax2+bx+c=0, a,b,c∈R have two roots in common. Then a+b is equal to

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The two equations $x^{3} + 1 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R$ have two roots in common. Then $a + b$ is equal to