Question

If the sum of the squares of zeroes of the quadratic polynomial f(x) = x^2-8x+p is 40, then find the value of p

Answer 1

We know that : Zero of a Quadratic Polynomial is the Root of the respective Quadratic Equation.

If 'α' and 'β' are Roots of a Quadratic Equation ax² + bx + c = 0, then :

✿  $\mathsf{Sum\;of\;the\;Roots\;(\alpha + \beta) = \frac{-b}{a}}$

✿  $\mathsf{Product\;of\;the\;Roots\;(\alpha \times \beta) = \frac{c}{a}}$

Given : Quadratic Polynomial : f(x) = x² - 8x + p

⇒ Quadratic Equation : x² - 8x + p = 0

Let 'Δ' and 'Ф' be the Roots of the Quadratic Equation x² - 8x + p = 0

⇒ Sum of the Roots (Δ + Ф) = 8

⇒ Product of the Roots (Δ × Ф) = p

Given : The Sum of the Squares of Zeroes of the Quadratic Polynomial f(x) = x² - 8x + p = 0 is 40

⇒ The Sum of the Squares of the Roots of the Quadratic Equation x² - 8x + p = 0 should be 40

⇒ Δ² + Ф² = 40

We know that : (a + b)² = a² + b² + 2ab

⇒ a² + b² = (a + b)² - 2ab

⇒ (Δ + Ф)² - 2ΔФ = 40

⇒ 8² - 2(p) = 40

⇒ 2p = 64 - 40

⇒ 2p = 24

⇒ p = 12