Question

If 2 is the root of the quadratic equation x^2 + 3x + k = 0, then the product of the roots of the equation is:

Answer 1

Answer:

Product of roots = -10

Step-by-step explanation:

Given :

2 is the root of the quadratic equation x² + 3x + k = 0

To find :

the product of the roots of the equation

Solution :

Let p(x) = x² + 3x + k

2 is a root of the given equation.

The result is zero when we substitute x = 2.

Put x = 2,

2² + 3(2) + k = 0

4 + 6 + k = 0

 10 + k = 0

  k = -10

∴ The value of k is -10

The relation between the roots and the coefficients is given as :

• Sum of roots = -(x coefficient)/x² coefficient

• Product of roots = constant term/x² coefficient

So,

⇒ product of roots = k/1

⇒ product of roots = k

⇒ product of roots  = -10

Answer 2

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.

ToFind :—

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,

$: \implies \rm \alpha \beta$

but ,

$\begin{gathered}{\large{\longrightarrow{\blue{\boxed{\green{\star{\rm{ \alpha \beta = \dfrac{c}{a}}}}}}}}} \\ \end{gathered}$

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

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

$\begin{gathered} : \implies \rm \alpha \beta = ( k - 10 )\\ \end{gathered}$

$\begin{gathered} : \implies \rm -2 = ( k - 10 )\\ \end{gathered}$

$\begin{gathered} : \implies \rm -2 + 10 = k \\ \end{gathered}$

$\begin{gathered} \therefore \rm k = 8 \\ \end{gathered}$

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

Hence , the quadratic equation is

➪ x² - 3x + 8 - 10 = 0

➪ x² - 3x - 2 = 0 ____________________________________

$\begin{gathered}{\large{\bold{\purple{\underline{\green{\star{\rm{ Verification\:of\:the\:Solution :}}}}}}}} \\ \end{gathered}$

The quadratic equation is :—

➪ x² - 3x - 2 = 0

Now,

Compare this equation with

➪ ax² + bx + c = 0

Therefore,

➪ a = 1 ,

➪ b= -3,

➪ c = -2

Le$\alpha \: and \: \beta$

be the roots of given quadratic equation.

We know,

$\begin{gathered}{\longrightarrow{ \blue{ \underline{\green{ \star {\rm Product\:of\:roots = \dfrac{c}{a}}}}}}} \\ \end{gathered}$

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

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

$\begin{gathered} \implies \rm \alpha \beta = -2 \\ \end{gathered}$

Therefore ,

the product of roots is -2 .

${\large{\underline{\rm{\purple{Hence, \: Verified } } } } }$

______________________________________

$\begin{gathered}{\large{\bold{\green{\underline{\blue{\star{\rm{ Formula\:Used :}}}}}}}} \\ \end{gathered}$

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

@Vikramjeeth