Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. ${{x}^{2}}-2x-8$.

Answer 1

Hint: Equate the given quadratic equation to zero and factorize it into linear terms by splitting the middle term method. Equate the linear terms to zero to calculate the roots of the equation. Use the fact that the sum and product of roots of the equation of the form $a{{x}^{2}}+bx+c$ are given by $\dfrac{-b}{a}$ and $\dfrac{c}{a}$ respectively to verify the relationship between the zeroes and the coefficients.

Complete step-by-step solution -
We have to calculate the roots of the equation ${{x}^{2}}-2x-8=0$ and verify the relationship between the zeroes and the coefficients.
We will factorize the given equation into linear terms by splitting the middle term method.
Thus, we can rewrite the given equation as ${{x}^{2}}-4x+2x-8=0$.
Taking out the common terms, we can rewrite it as $x\left( x-4 \right)+2\left( x-4 \right)=0$.
We can further write the above equation as $\left( x-4 \right)\left( x+2 \right)=0$.
Thus, we have $x-4=0$ or $x+2=0$.
So, the roots of the equation ${{x}^{2}}-2x-8=0$ are $x=4$ and $x=-2$.
We will now verify the relationship between the zeroes and the coefficients of the equation ${{x}^{2}}-2x-8=0$.
We know that the sum and product of roots of the equation of the form $a{{x}^{2}}+bx+c$ are given by $\dfrac{-b}{a}$ and $\dfrac{c}{a}$ respectively.
Substituting $a=1,b=-2,c=-8$ in the above expression, the sum, and product of roots of the equation ${{x}^{2}}-2x-8=0$ is $\dfrac{-\left( -2 \right)}{1}=2$ and $\dfrac{-8}{1}=-8$.
We know that the roots of the equation ${{x}^{2}}-2x-8=0$ are $x=4$ and $x=-2$.
Thus, the sum of roots is $=4+\left( -2 \right)=2$ and the product of roots is $=4\times \left( -2 \right)=-8$. So, the relation between the zeroes and the coefficients of the equation ${{x}^{2}}-2x-8=0$ holds true.
Hence, the roots of the equation ${{x}^{2}}-2x-8=0$ are $x=4$ and $x=-2$.

Note: We can also solve this question by factorize the quadratic polynomial by completing the square method or calculating the discriminant method and then verify the relation between the roots and the coefficients of the quadratic equation.