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Relation between Roots and Coefficients for Quadratic

If α, β are t...

Question

If $α, β$ are the roots of $3 x^{2} - 2 x + 1 = 0$ , then the equation whose roots are $- \frac{α}{3}, - \frac{β}{3}$ is

27 x^{2} + 6 x - 1 = 0

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9 x^{2} + 6 x + 1 = 0

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27 x^{2} + 6 x + 1 = 0

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27 x^{2} - 6 x + 1 = 0

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Solution

The correct option is C $27 x^{2} + 6 x + 1 = 0$
$f (x) = 3 x^{2} - 2 x + 1$
$f (x) = 0 \to α, β$ are the roots.
The equation whose roots are $- \frac{α}{3}, - \frac{β}{3}$ is
$f (- 3 x) = 0$
$\Rightarrow 3 (- 3 x)^{2} - 2 (- 3 x) + 1 = 0 \Rightarrow 27 x^{2} + 6 x + 1 = 0$

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If α,β are the roots of 3x2−2x+1=0, then the equation whose roots are −α3,−β3 is

The correct option is C 27x2+6x+1=0f(x)=3x2−2x+1 f(x)=0→α,β are the roots. The equation whose roots are −α3,−β3 is f(−3x)=0 ⇒3(−3x)2−2(−3x)+1=0⇒27x2+6x+1=0

If $α, β$ are the roots of $3 x^{2} - 2 x + 1 = 0$ , then the equation whose roots are $- \frac{α}{3}, - \frac{β}{3}$ is

The correct option is C $27 x^{2} + 6 x + 1 = 0$
$f (x) = 3 x^{2} - 2 x + 1$
$f (x) = 0 \to α, β$ are the roots.
The equation whose roots are $- \frac{α}{3}, - \frac{β}{3}$ is
$f (- 3 x) = 0$
$\Rightarrow 3 (- 3 x)^{2} - 2 (- 3 x) + 1 = 0 \Rightarrow 27 x^{2} + 6 x + 1 = 0$