Question

21. If not root of the quadratic equation 2x + x - 6 = 0 is 2, find the value of k. Also, find the other root​

Answer 1

$\underline{\bf{Correct \: Question \: :-}}$

If root of the quadratic equation 2x² + kx - 6 = 0 is 2, find the value of k. Also, find the other root.

Methods for doing such questions :

$\bullet$By Using Splitting Middle Term Method

$\bullet$By Using Quadratic Formula

Now, Firstly

Steps for Splitting Middle Term Method :

★ ➤ We have to split the middle term of equation, so that their sum is equal to the middle term and their product is equal to the product of coefficient of x² and constant term.

Let's solve it :

$\dashrightarrow\:\:\sf 2 \times ( 2 )^{2} + k × 2 - 6 = 0$

$\dashrightarrow\:\:\sf 8 + 2k - 6 = 0$

$\dashrightarrow\:\:\sf 2 + 2k = 0$

$\dashrightarrow\:\:\sf k = \dfrac{-2}{2}$

$\dashrightarrow\:\:\sf k = - 1$

$\boxed{ \bf{k = - 1}}$

Now , By using Splitting method :

$\rightarrow\:\:\sf 2x^{2} - x - 6 = 0$

$\rightarrow\:\:\sf 2x^{2} - ( 4 - 3 )x - 6 = 0$

$\rightarrow\:\:\sf 2x^{2} - 4x + 3x - 6 = 0$

$\rightarrow\:\:\sf 2x ( x - 2 ) + 3 ( x - 2 ) = 0$

$\rightarrow\:\:\sf x - 2 = 0$

$\rightarrow\:\:\sf x = 2$

or

$\rightarrow\:\:\sf 2x + 3 = 0$

$\rightarrow\:\:\sf x = \dfrac{-3}{2}$

Hence, The other root is :

$\boxed{\therefore \sf{x = \dfrac{-3}{2} }}$

____________________________

Alternate Method :

$\large\mathtt{|| \: Quadratic \; \; Formula \: ||}$

$\;\tt{\rightarrow\;\; \dfrac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Here,

• 2x² - x - 6 = 0

By comparing it with the general form of quadratic equation which is ,

• $\boxed{\boxed{\sf{ax^{2} + bx + c = 0}}}$

⠀⠀$\dag$ a = 2

⠀⠀$\dag$ b = (-1)

⠀⠀$\dag$ c = (-6)

Finding the Discriminant first, which is nothing but :

$\bigstar {\sf { D = b^{2} - 4 ac}}$

Substituting the respective values here, we get :-

$\implies\sf D = b^{2} - 4 ac$

$\implies\sf D = (-1)^{2} - 4 \times 2 \times (-6)$

$\implies\sf D = 1 - 8 \times (-6)$

$\implies\sf D = 1 + 48$

$\rightarrow$ D = 49

Plugging the values now ,

$\;\sf{\rightarrow\;\; \dfrac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

$\;\sf{\rightarrow\;\; \dfrac{- (-1) \:\pm\:\sqrt{49}}{2(2)}}$

$\;\sf{\rightarrow\;\; \dfrac{1 \:\pm\:\sqrt{49}}{4}}$

$\;\sf{\rightarrow\;\; \dfrac{1 \:\pm\:{7}}{4}}$

Now,

$\;\sf{\rightarrow\;\; \dfrac{1 \: + \:{7}}{4}}$

$\;\sf{\rightarrow\;\; \dfrac{{8}}{4}}$

$\;\sf{\rightarrow\;\; 2 }$

Then,

$\;\sf{\rightarrow\;\; \dfrac{1 \: - \:{7}}{4}}$

$\;\sf{\rightarrow\;\; \dfrac{{-6}}{4}}$

$\;\sf{\rightarrow\;\; \dfrac{{-3}}{2}}$

Hence, The other root is :

$\boxed{\therefore \sf{x = \dfrac{-3}{2} }}$

_________________________

$\underline{\underline{\maltese\:\: \textbf{\textsf{Quadratic \: Equation}}}}$

$\bigstar$ If p(x) is a quadratic polynomial , then p(x) = 0 is called a quadratic equation .

$\underline{\underline{\maltese\:\: \textbf{\textsf{Roots \: of \: Quadratic \: Equation}}}}$

$\bigstar$Let p(x) = 0 be a quadratic equation , then the zeros of the polynomial p(x) are called the roots of the equation p(x) = 0.

⠀⠀$\underline{\underline{\maltese\:\: \textbf{\textsf{General \: form \: of \: Quadratic \: Equation}}}}$

⠀⠀⠀⠀⠀⠀$\underline{\bf{ax^{2} + bx + c = 0}}$

Where,

$\bullet$a , b and c are real numbers respectively.

_______________________

Answer 2

Correct Question:

If one root of the quadratic equation 2x² + kx - 6 = 0 is 2, find the value of k. Also find the other root.

Solution:

Let us assume that the other root of the given quadratic equation is m

then two roots of the equation are: 2 and m

Comparing 2x² + x - 6 = 0 with ax² + bx + c = 0

we will get, a = 2, b = k, c = -6

Now we know the relation between roots and coefficient of the quadratic equation, so

Sum of roots = 2 + m = -b/a = -k/2

so, 4 + 2m = -k ___equation(1)

also,

product of roots = 2m = c/a = -6/2 = -3

so, 2m = -3 ___equation(2)

putting value of 2m from equation(2) into equation(1)

➡ 4 + 2m = -k

➡ 4 - 3 = -k

➡ k = -1

and putting value of k in equation(1)

➡ 4 + 2m = -k

➡ 4 + 2m = -(-1)

➡ 4 + 2m = 1

➡ 2m = -3

➡ m = -3/2

so, value of k is -1 and other root is -3/2.