Question

find the zeroes of verify the relationship 2x^2-3+5x.​

Answer 1

full question :- find the zeroes of the polynomial 2x² - 3 + 5x and verify the relationship between it's zeroes and coefficients.

first of all, we've to write the polynomial in it's standard form which is ax² + bx + c

= 2x² + 5x - 3

let us find it's roots now by splitting the middle term.

= 2x² + (6x - x) - 3

= 2x² + 6x - x - 3

= 2x(x + 3) - 1(x + 3)

= (x + 3) (2x - 1)

equating both the factors with 0

• x + 3 = 0

➡ x = -3

• 2x - 1 = 0

➡ 2x = 1

➡ x = 1/2

now, relationship between it's zeroes and coefficients

sum of zeroes = -3 + 1/2

= -6/2 + 1/2

= -5/2

also -b/a = -5/2

hence verified.

product of zeroes = -3 × 1/2

= -3/2

also c/a = -3/2

hence verified.

Answer 2

Step-by-step explanation:

first of all, we've to write the polynomial in it's standard form which is $\large \:ax^{2} \:+ \:bx \:+ \:c$

$\large \:= \:2x^{2} \:+ \:5x \:- \:3$

let us find it's roots now by splitting the middle term.

$\large \:= \:2x^{2} \:+ \:( \:6x \:- \:x \:) \:- \:3$

$\large \:= \:2x^{2} \:+ \:6x \:- \:x \:- \:3$

$\large \:= \:2x \:( \:x \:+ \:3 \:) \:- \:1 \:( \:x \:+ \:3 \:)$

$\large \:= \:( \:x \:+ \:3 \:) \:( \:2x \:- \:1 \:)$

equating both the factors with 0

$\large \:x \:+ \:3 \:= \:0$

$\large \:: \:x \:= \:-3$

$\large \:2x \:- \:1 \:= \:0$

$\large \:: \:2x \:= \:1$

$\large \:: \:x \:= \frac{1}{2}$

now, relationship between it's zeroes and coefficients

sum of zeroes $\large \:= \:-3 \:+ \frac{1}{2}$

$\large \:= \frac{-6}{2} \:+ \frac{1}{2}$

$\large \:= \frac{-5}{2}$

also $\large \frac{-b}{a} \:= \frac{-5}{2}$

hence verified.

product of zeroes $\large \:= \:-3 \:× \frac{1}{2}$

$\large \:= \frac{-3}{2}$

also $\large \frac{c}{a} \:= \frac{-3}{2}$

hence verified