Let - be the roots of the quadratic equation 1-mx2-2-3mx-3m-0- m - has distinct integer roots- Then find the equation whose roots are -5 - -5-

Given: (1 m)x^2 (2 3m)x 3m=0 On comparing with standard form of nbsp; quadratic equation a nbsp; x^2+ b x+c=0 we get, a=(1 m) , b= (2 3m) , c= 3m If roots ...

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Standard XII

Mathematics

Transformation of Roots: Linear Combination of Roots

Let α,β be th...

Question

Let $α, β$ be the roots of the quadratic equation $(1 - m) x^{2} - (2 - 3 m) x - 3 m = 0, m \in W$ has distinct integer roots. Then find the equation whose roots are $α + 5 & β + 5$ .

x^{2} - 12 x - 35 = 0

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x^{2} - 12 x + 35 = 0

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x^{2} + 12 x - 35 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

x^{2} + 12 x + 35 = 0

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Solution

The correct option is B $x^{2} - 12 x + 35 = 0$
Given: $(1 - m) x^{2} - (2 - 3 m) x - 3 m = 0$
On comparing with standard form of quadratic equation $a x^{2} + b x + c = 0$
we get, $a = (1 - m), b = - (2 - 3 m), c = - 3 m$

If roots are distinct integers, then discriminant should be perfect square of an integer.
$D = b^{2} - 4 a c = (- (2 - 3 m))^{2} - 4 \cdot (1 - m) \cdot (- 3 m)$
$\Rightarrow D = 4 + 9 m^{2} - 12 m + 12 m - 12 m^{2}$
$\Rightarrow D = - 3 m^{2} + 4$
Also, $D > 0 \Rightarrow - 3 m^{2} + 4 > 0$
$\Rightarrow 3 m^{2} - 4 < 0$
$\Rightarrow (\sqrt{3} m - 2) (\sqrt{3} m + 2) < 0$
$\Rightarrow m \in (\frac{- 2}{\sqrt{3}}, \frac{2}{\sqrt{3}})$

Since, $m \in W \Rightarrow m \in {0, 1}$
If $m = 0, D = - 3 (0)^{2} + 4 = 4,$ which is a perfect square.
Similarly, if $m = 1, D = - 3 (1)^{2} + 4 = 1$ which is a perfect square.

Hence, the possible values of $m$ are $0, 1$ .
When $m = 0$ , the quadratic equation becomes $x^{2} - 2 x = 0$
$\Rightarrow α = 0; β = 2 \dots (i)$

When $m = 1$ , the equation becomes $x - 3 = 0$ which is not a quadratic equation since coefficient of $x^{2}$ is $0$

Thus, we have $α = 0; β = 2$

Now, we need to find the equation whose roots are $α + 5 & β + 5$
$∴ roots are 0 + 5 = 5, 2 + 5 = 7$
$∴ The equation is (x - 5) (x - 7) = 0$
$\Rightarrow x^{2} - 12 x + 35 = 0$

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Let α,β be the roots of the quadratic equation (1−m)x2−(2−3m)x−3m=0,m∈W has distinct integer roots. Then find the equation whose roots are α+5 & β+5.

Let $α, β$ be the roots of the quadratic equation $(1 - m) x^{2} - (2 - 3 m) x - 3 m = 0, m \in W$ has distinct integer roots. Then find the equation whose roots are $α + 5 & β + 5$ .