Question

Find the zeroes of the following quadratic
polynomial and verify the relationship
between the zeroes and the coefficients x2-2x-8

Answer 1

Answer :-

• Zeroes = 4 and -2

To Find :-

• Zeroes of the polynomial x² - 2x - 8

Given :-

• Polynomial x² - 2x - 8

Step By Step Solution :-

We can easily find the zeroes of the given polynomial by splitting the middle term.

So let's do it !!

$\implies \sf {x}^{2} - 2x - 8 \\ \\ \implies \sf {x}^{2} -4x + 2x - 8 \\ \\ \implies \sf \: x(x - 4) + 2(x - 4) \\ \\ \implies \sf (x - 4)(x + 2) \\ \\ \sf \:Zeroes \downarrow\\ \\ \pink{ \sf x - 4 = 0 \implies \: x = 4 }\\ \\ \green {\sf \: x + 2 = 0 \implies \: x = - 2}$

Now α = 4 and β = -2

Relationship ⤵

As we know,

If α and β are zeroes of the quadratic polynomial ax² + bx + c, then ⤵

$\bigstar\boxed{ \underline{ \mathfrak{ \purple{\alpha + \beta = \cfrac{ - b}{a} }}}}$

$\bigstar\boxed{ \underline{ \mathfrak{ \red{ \alpha \beta = \cfrac{c}{a}}}}}$

By substituting the values in 1st eq. ⤵

$\implies \sf4 + ( - 2) = \cfrac{ - ( - 2)}{1} \\ \\ \implies \sf 2 = 2$

1st relationship verified.

Now substituting the values in 2nd eq. ⤵

$\implies \sf4 \times ( - 2) = \cfrac{ - 8}{1} \\ \\ \implies \sf - 8 = - 8$

2nd relationship verified.

Therefore, 4 and -2 are zeroes of the polynomial x² - 2x - 8

__________________________