Question

If α and β are the zeroes of the polynomial x2 + 5x + 8, then the value of α2 + β2 is……………

Answer 1

$\huge\underline{ \mathbb\red{⭐A} \green{n} \mathbb\blue{S} \purple{w} \mathbb \orange{E} \pink{r}} \:$

➡️Alpha and beta are zeros of the polynomial f(x)=x^2-5x+k.

➡️let alpha=a &beta=b

➡️a+b=-(-5/1)=5;ab=k/1=k

➡️Now, a-b=1

➡️squaring both sides

➡️(a-b)^2=1

➡️(a+b)^2-4ab=1

➡️25-4k=1

➡️4k=24

➡️k=6.

Answer 2

$\Large{\underline{\underline{\mathfrak{\red{\bf{Solution}}}}}}$

$\Large{\underline{\mathfrak{\orange{\bf{Given}}}}}$

• Equation, x² + 5x + 8 = 0
• α and β are the zeroes of the polynomial

$\Large{\underline{\mathfrak{\orange{\bf{Find}}}}}$

• Value of α² + β²

$\Large{\underline{\underline{\mathfrak{\red{\bf{Explanation}}}}}}$

Using Formula

$\small{\boxed{\tt{\green{\:Sum\:of\:zeroes\:=\:\dfrac{-(coefficient\:of\:x)}{(coefficient\:of\:x^2)}}}}}$

$\small{\boxed{\tt{\blue{\:product\:of\:zores\:=\:\dfrac{(Constant\:part)}{(coefficient\:of\:x^2)}}}}}$

Now,

➡ Sum of zeroes = -(5)/1

➡ α + β = -5 __________(1)

Again,

➡Product of zeroes = 8/1

➡ α . β = 8 ___________(2)

Using Formula,

$\small{\boxed{\tt{\green{\:( \alpha+ \beta)^2\:=\:\alpha^2\:+\:\beta^2\:+\:2\alpha\:\beta}}}}$

Keep above all values,

➡ (-5)² = α² + β² + 2 × 8

➡ 25 = α² + β² + 16

➡ α² + β² = 25 - 16

➡ α² + β² = 9

$\Large{\underline{\underline{\mathfrak{\red{\bf{Hence}}}}}}$

• Value of α² + β² = 9

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