Question

if zeroes of polynomial X2+px+q are double in value to tha zeroes of 2x2-5x-3, find the value of p and q?​

Answer 1

Answer:

☑ let the given equations are x^2 + px + q and 2x^2 - 5x - 3

☑ first of all find the zeroes of the equation 2x^2 - 5x - 3 equate this equation with zero

2x^2 - 5x - 3 = 0

⟹ 2x^2 - 6x + x - 3 = 0

⟹ 2x ( x - 3 ) + 1 ( x - 3 )= 0

⟹ ( 2x + 1 ) ( x - 3) = 0

⟹ x = -1 /2

OR

⟹ x = 3

☑ let the given values of x, take it as m and n as a zeros, so

⟹m = -1/2

⟹n = 3

Now,

consider the zeros of the equation (x^2 + px + q ) are α and β

consider the zeros of the equation (x^2 + px + q ) are α and β compare this equation with standard form (ax^2 + bx + c = 0)

therefor,

a = 1

b = p

c = q

☑ By the given condition, zeroes of polynomial X2+px+q are double in value to tha zeroes of 2x2-5x-3,

α = 2 m

β = 2 n

therefor α= 2 x ( -1 / 2) = -1

and β= 2 x ( 3) = 6

☑ we know, the relation of the constant with roots of equation is

α + β = - b / a

and

α.β = c / a

let, α.β = c / a

⟹ (-1).6 = q / 1

⟹ ( -6) = q

so, q = -6,

$\boxed{\textbf{\large{q=-6}}}$

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☑ Now, the value of p is

α + β = (-b) / a

⟹(-1)+ 6 = (-p) / 1

⟹ ( 5) = (-p)

p = -5

$\boxed{\textbf{\large{p=-5}}}$

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