Question

If one root of the quadratic equation $ 2{x^2} + kx - 6 = 0 $ is $ 2 $ , find the value of $ k $ . Also, find the other root.

Answer 1

Hint:As we know that the above equation is a quadratic equation. We know that the quadratic equations are considered a polynomial equation of degree $ 2 $ in one variable of the form $ a{x^2} + bx + c = 0 $ . It is also the general expression of the quadratic equation. The values of $ x $ satisfying the quadratic equation are known as the roots of the quadratic equation. We can find the value of other roots by using the sum and the product formula of the roots.

Complete step by step solution:
As per the given question we have an equation $ 2{x^2} + kx - 6 = 0 $ and one of the roots is $ 2 $ .
We will find the other root by the formula of sum and the product. We know that if $ \alpha $ and $ \beta $ are the two roots of the quadratic equation then the sum of the roots of the equation is
 $ \alpha + \beta = \dfrac{{ - b}}{a} $ and the product of $ \alpha \beta = \dfrac{c}{a} $ .
We have given one root of the equation by putting that value n the equation we get:
$ 2{x^2} + kx - 6 = 0\\
\Rightarrow 2{(2)^2} + k \times 2 - 6 = 0 $ .
On further solving we have,
 $ \Rightarrow 2k + 8 - 6 = 0\\
\Rightarrow 2k + 2 = 0 $ ,
 It gives $ k = - 1 $ .
By putting the value of $ k $ in the equation we have :
 $ 2{x^2} + ( - 1)x - 6 = 0 $ .
So our new equation is $ 2{x^2} - x - 6 = 0 $ .
In the quadratic equation we have
 $ a = 2,b = - 1,c = - 6 $ .
Now using the sum of the roots of the quadratic equation;
 $ \alpha + \beta = \dfrac{{ - b}}{a} $ ,
 by putting the values we get:
  $ 2 + \beta = \dfrac{{ - ( - 1)}}{2} $
 $ 2(2 + \beta ) = 1 \\
\Rightarrow 4 + 2\beta = 1 $ .
It gives us $ \beta = \dfrac{{ - 3}}{2} $ .
Hence the other root is $ \dfrac{{ - 3}}{2} $ .

Note: Before solving this kind of question we should have proper knowledge of quadratic equations and their roots formula. We can also solve this question by using the discriminant formula. Also we can find the roots by splitting the terms and then find the factors if we do not know the formula of sum and product of roots.