Question

If sum of the squares of zeroes of the quadratic polynomial f(x)

= x^2 – 8x + k is 40, the value

of k is :

​

Answer 1

$\huge{\underline{\bf{\blue{Question:-}}}}$

If sum of squares of zeroes of a quadratic polynomial p(x)= x² - 8x + k is 40 ,then , find the value of k.

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$\large{\underline{\bf{\pink{Answer:-}}}}$

k = 12

$\large{\underline{\bf{\purple{Explanation:-}}}}$

$\large{\underline{\bf{\green{Given:-}}}}$

A quadratic polynomial is given as:-

p(x) = x² - 8x + k

$\large{\underline{\bf{\green{To\:Find:-}}}}$

we need to find the value of k.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let α and β are the zeroes of the quadratic polynomial then ,

sum of zeroes (α+β)

$:\implies\bf\:\frac{- coeff.\:of\:x}{ coefficient\:of\:{x}^{2}}$

$:\implies\bf\frac{-(-8)}{1}=8$

and,

product of zeroes (αβ)

$:\implies\bf\:\frac{constant\:term}{ coefficient\:of\:{x}^{2}}$

$:\implies\bf\frac{k}{1}=k$

Now,

➥ α² + β² = (α + β)²- 2αβ

➜ 40 = 8² - 2k

➜ 40 = 64 - 2k

➜ 2k = 24

➥ k = 12

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Answer 2

Gɪᴠᴇɴ :-

• Quadratic Equation = x² - 8x + k = 0
• sum of Square of zeros = 40 .

Tᴏ Fɪɴᴅ :-

• value of k ?

ᴄᴏɴᴄᴇᴘᴛ ᴜsᴇᴅ :-

→ The sum of the roots of the Equation ax² + bx + c = 0 , is given by = (-b/a)

and ,

→ Product of roots of the Equation is given by = c/a.

Sᴏʟᴜᴛɪᴏɴ :-

Let us Assume That, The roots of the given Quadratic Polynomial are ɑ & β .

Now,

Comparing The given Equation with ax² + bx + c = 0 , we get,

→ a = 1

→ b = (-8)

→ c = k

So,

→ ɑ + β = (-b/a) = -(-8)/1 = 8 ----- Equation(1)

→ ɑ × β = c/a = k/1 = k ------ Equation(2)

Also , we have given that,

→ ɑ² + β² = 40 ------ Equation (3).

__________

Now, Taking Equation (1) ,

→ ɑ + β = 8

Squaring Both sides we get,

→ (ɑ + β)² = 8²

→ ɑ² + β² + 2*ɑ*β = 64

Putting value of Equation (2) & (3) in LHS now, we get,

→ 40 + 2k = 64

→ 2k = 64 - 40

→ 2k = 24

→ k = 12 (Ans.)

Hence, value of k will be 12.