Question

find the equation that formed by increasing each root of 2x²-3x-1=0by 1​

Answer 1

SOLUTION

TO DETERMINE

The quadratic equation that is formed by increasing each root of 2x²- 3x - 1 = 0 by 1

EVALUATION

Here the given Quadratic equation is

$\sf{2 {x}^{2} - 3x - 1 = 0 \: \: \: \: .....(1) }$

Let $\sf{ \alpha \: \: and \: \beta }$

are the roots of the quadratic equation (1)

$\displaystyle\sf{ \alpha + \beta = \frac{3}{2} \: \: and \: \: \alpha \beta = - \frac{1}{2} }$

Now we have to find the quadratic equation whose roots are

$\sf{ \alpha + 1 \: \: and \: \: \beta + 1}$

Now

$\sf{( \alpha + 1) + ( \beta + 1)}$

$\sf{ = ( \alpha + \beta + 2)}$

$\displaystyle \sf{ = \frac{3}{2} + 2 }$

$\displaystyle \sf{ = \frac{7}{2} }$

Again

$\displaystyle \sf{ ( \alpha + 1)( \beta + 1) }$

$\displaystyle \sf{ = \alpha \beta + \alpha + \beta + 1 }$

$\displaystyle \sf{ = - \frac{1}{2} + \frac{3}{2} + 1 }$

$\displaystyle \sf{ = 1 + 1 }$

$\displaystyle \sf{ = 2}$

Hence the required Quadratic equation is

$\displaystyle \sf{ {x}^{2} - \frac{7x}{2} + 2 = 0 }$

$\displaystyle \sf{ \implies 2{x}^{2} - 7x + 4 = 0 }$

FINAL ANSWER

The required Quadratic equation is

$\displaystyle \sf{ 2{x}^{2} - 7x + 4 = 0 }$

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