Question

Find the zeroes of the quadratic polynomial Z^2 – 2Z– 8 and verify the relationship between the zeroes and the coefficients.


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Answer 1

$\star\:\:\:\bf\large\underline\blue{Question:-}$

• Find the zeros of the Quadratic equation and verify the relationship between roots and coefficients.

$\star\:\:\:\bf\large\underline\blue{Solution:-}$

${z}^{2} - 2z - 8 = 0$

$\implies {z}^{2} - 4z + 2z - 8 = 0$

$\implies z(z - 4) + 2(z - 4) = 0$

$\implies( z+ 2)(z - 4) = 0$

Therefore, Either $(z+2)=0$ or $(z-4) =0$.

So,

$z + 2 = 0$

$\implies z = - 2$

$z - 4 = 0$

$\implies z = 4$

Hence,

$\alpha = - 2$

$\beta = 4$

$\star\:\:\:\bf\large\underline\blue{Verification:-}$

• Relationship between roots and coefficients are given by the formula

$\alpha + \beta = \frac{ - b}{a}$

$\alpha \beta = \frac{c}{a}$

Where a, b and c are the coefficients of the equation.

Now,

$\alpha + \beta = - 2 + 4 = 2$

Also,

$\frac{ - b}{a} = \frac{ - ( - 2)}{1} = 2$

So,

$\alpha + \beta = \frac{ - b}{a}$

Now,

$\alpha \beta = 4 \times ( - 2) = - 8$

Also,

$\frac{c}{a} = \frac{ - 8}{1} = - 8$

Hence,

$\alpha \beta = \frac{c}{a}$

Verified.

$\star\:\:\:\bf\large\underline\blue{Answer:-}$

• Roots are -2 and 4.
• Relationships are verified.