Question

If the product of the roots of the equation x2 - 3x + k = 10 is -2 then the value of k is

Answer 1

Question:

If the product of the roots of the equation x² - 3x + k = 10 is -2 then Find the value of k.

Answer:

Value of k is :

Given:

➛The equation is x² - 3x + k = 10.

➛ Product of roots of the equation is -2.

To Find:

The value of k.

Solution:

We are given,

➛The equation is x² - 3x + k = 10.

➛ Product of roots of the equation is -2.

The given quadratic equation can also be written as

➞ x² - 3x + k - 10 = 0

NOW,

compare the given quadratic equation with ax² + bx + c = 0 .

we get

a = 1 ,

b = -3 ,

c = k - 10

Let $\alpha \: and \: \beta$ be the roots of given quadratic equation.

Therefore,

According to given condition,

$: \implies \rm \alpha \beta = -2$

but,

${\large{\longrightarrow{\blue{\boxed{\red{\star{\rm{ \alpha \beta = \dfrac{c}{a}}}}}}}}}$

$: \implies \rm \alpha \beta = \dfrac{c}{a}$

$: \implies \rm \alpha \beta = \dfrac{( k - 10 )}{1}$

$: \implies \rm \alpha \beta = ( k - 10 )$

$: \implies \rm -2 = ( k - 10 )$

$: \implies \rm -2 + 10 = k$

$\therefore \rm k = 8$

${\large{\therefore{\orange{\boxed{\green{\star{\rm{ \alpha \beta = \dfrac{c}{a}}}}}}}}}$

Hence , the quadratic equation is

➪ x² - 3x + 8 - 10 = 0

➪ x² - 3x - 2 = 0

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${\large{\bold{\purple{\underline{\blue{\star{\rm{ Verification\:of\:the\:Solution :}}}}}}}}$

The quadratic equation is :

x² - 3x - 2 = 0

Now,

Compare this equation with

ax² + bx + c = 0

Therefore,

a = 1 ,

b= -3,

c = -2

Let $\alpha \: and \: \beta$ be the roots of given quadratic equation.

We know,

${\longrightarrow{ \red{ \underline{\pink{ \star {\rm Product\:of\:roots = \dfrac{c}{a}}}}}}}$

$\implies \rm \alpha \beta = \dfrac{c}{a}$

$\implies \rm \alpha \beta = \dfrac{-2}{1}$

$\implies \rm \alpha \beta = -2$

Therefore , the product of roots is -2 .

${\large{\underline{\rm{\purple{Hence, \: Verified<em>}</em><em>}</em><em>}</em><em>}</em><em>}</em>$

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${\large{\bold{\blue{\underline{\red{\star{\rm{ Formula\:Used :}}}}}}}}$

${\large{ \blue{ \boxed{\boxed{\pink{ \star {\rm Product\:of\:roots = \dfrac{c}{a}}}}}}}}$