Question

Find the zeros of the given quadratic polynomial p(x) =x^2-2x-8​

Answer 1

Required Answer:-

Given:

• p(x) = x² - 2x - 8

To Find:

• The zeros of the given equation.

Solution:

Given,

→ p(x) = x² - 2x - 8

→ x² - 2x - 8 = 0

→ x² - (4 - 2)x - 8 = 0

→ x² - 4x + 2x - 8 = 0

→ x(x - 4) + 2(x - 4) = 0

→ (x + 2)(x - 4) = 0

By Zero-Product rule,

→ (x + 2) = 0 or (x - 4) = 0

→ x = -2, 4

★ So, the zeros of the given equation are -2 and 4.

Answer:

• x = -2, 4.

•••♪

Answer 2

Required Answer :

• The value of x = -2 or 4.

Concept :

☆ The given quadratic equation can be solved by two ways :-

• By Prime factorisation - We'll make the common factors of the middle term and then take a number as common to find the value of x. This method is known as splitting the middle term.
• By quadratic formula - In this formula, we'll find out the discriminant of the equation using formula b²-4ac, and then put it in the another formula to obtain the value of x. This method is known as quadratic formula.

Step by step explanation :

Given :

• Quadratic equation = x²-2x-8

To find :

• Roots/zeroes of the quadratic equation.

Solution :

⠀⠀⠀⠀⠀⠀⠀-------------First way-----------

❥ By Prime factorisation,

$p(x) = {x}^{2} - 2x - 8$

$→$ $p(x) = {x}^{2} - 4x + 2x - 8$

• Taking x and (+2) as common.

$→$ $p(x) = x(x - 4) + 2(x - 4)$

$→$ $(x - 4)(x + 2)$

• Now, for finding zeroes, put p(x) = 0

∴ $p(x) = 0$

$→$ $(x - 4)(x + 2) = 0$

$→$ $(x - 4) = 0 \: or \: (x + 2) = 0$

$→$ $x = 4 \: or \: x = - 2$ ✔️

⠀⠀ ⠀⠀-----------------Second way--------------

❥By quadratic formula,

Given, $p(x) = {x}^{2} - 2x - 8$

• On comparing p(x) with ax²+bx+c, we get

1. a = 1
2. b = -2
3. c = -8

• Finding Discriminant using the formula, b²-4ac.

${b}^{2} - 4ac$

$→$ $( - 2)^{2} - 4( 1)( - 8)$

$→$ $4 + 32$

$→$ $36$ ------(i)

• Now, using the quadratic formula including the value of discriminant for finding the roots.

$\frac{ - b± \sqrt{b^{2} - 4ac } }{2a}$

$→$ $\frac{ +2 ± \sqrt{36 } }{2}$ [ from (i) ]

$→$ $\frac{2 - 6}{2} \: or \: \frac{2 + 6}{2}$

$→$ $\frac{ - 4}{2} \: or \: \frac{8}{2}$

∴$∴ \: x = - 2 \: or \: 4$ ✔️