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Question: If one root of the quadratic equation \( 2{x^2} + kx - 6 = 0 \) is \( 2 \) , find the value of \( k ...

If one root of the quadratic equation 2x2+kx−6=02{x^2} + kx - 6 = 0 is 22 , find the value of kk . Also, find the other root.

Explanation

Solution

As we know that the above equation is a quadratic equation. We know that the quadratic equations are considered a polynomial equation of degree 22 in one variable of the form ax2+bx+c=0a{x^2} + bx + c = 0 . It is also the general expression of the quadratic equation. The values of xx 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 2x2+kx−6=02{x^2} + kx - 6 = 0 and one of the roots is 22 .
We will find the other root by the formula of sum and the product. We know that if α\alpha and β\beta are the two roots of the quadratic equation then the sum of the roots of the equation is
α+β=−ba\alpha + \beta = \dfrac{{ - b}}{a} and the product of αβ=ca\alpha \beta = \dfrac{c}{a} .
We have given one root of the equation by putting that value n the equation we get:
2x2+kx−6=0 ⇒2(2)2+k×2−6=02{x^2} + kx - 6 = 0\\\ \Rightarrow 2{(2)^2} + k \times 2 - 6 = 0 .
On further solving we have,
⇒2k+8−6=0 ⇒2k+2=0\Rightarrow 2k + 8 - 6 = 0\\\ \Rightarrow 2k + 2 = 0 ,
It gives k=−1k = - 1 .
By putting the value of kk in the equation we have :
2x2+(−1)x−6=02{x^2} + ( - 1)x - 6 = 0 .
So our new equation is 2x2−x−6=02{x^2} - x - 6 = 0 .
In the quadratic equation we have
a=2,b=−1,c=−6a = 2,b = - 1,c = - 6 .
Now using the sum of the roots of the quadratic equation;
α+β=−ba\alpha + \beta = \dfrac{{ - b}}{a} ,
by putting the values we get:
2+β=−(−1)22 + \beta = \dfrac{{ - ( - 1)}}{2}
2(2+β)=1 ⇒4+2β=12(2 + \beta ) = 1 \\\ \Rightarrow 4 + 2\beta = 1 .
It gives us β=−32\beta = \dfrac{{ - 3}}{2} .
Hence the other root is −32\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.