If roots - - of the equations x 2- px -16-0 satisfy the relation -2-2-9- then write the value of P-

Since, α, β are the roots of the equation. x^2 px+16=0 ⇒α + β = b/a= ( p)/1=P and α β= c/a=16/1=16 Now, α^2 + β^2=9 ⇒ (α + β)^2 2 αβ = 9 ⇒ p^2 2 × 16 = 9 ⇒ ...

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Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

If roots α, β...

Question

If roots $α, β$ of the equations $x^{2} - p x + 16 = 0$ satisfy the relation $α^{2} + β^{2} = 9$ , then write the value of P.

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Solution

Since, $α, β$ are the roots of the equation. $x^{2} - p x + 16 = 0 \Rightarrow α + β = \frac{- b}{a} = \frac{- (- p)}{1} = P a n d α β = \frac{c}{a} = \frac{16}{1} = 16 N o w, α^{2} + β^{2} = 9 \Rightarrow (α + β)^{2} - 2 α β = 9 \Rightarrow p^{2} - 2 \times 16 = 9 \Rightarrow p^{2} 9 + 32 \Rightarrow p = \sqrt{41}$

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If roots α,β of the equations x2−px+16=0 satisfy the relation α2+β2=9, then write the value of P.

Since, α,β are the roots of the equation. x2−px+16=0⇒α+β=−ba=−(−p)1=Pand α β=ca=161=16Now,α2+β2=9⇒(α+β)2−2αβ=9 ⇒p2−2×16=9⇒p29+32⇒p=√41

If roots $α, β$ of the equations $x^{2} - p x + 16 = 0$ satisfy the relation $α^{2} + β^{2} = 9$ , then write the value of P.