Question

If the sum abd product of the zeroes of the polynomial ax^2-5x+c is equal to 10 eqch.find the values of a and c

Answer 1

Question

If sum and product of the zeros of the polynomial ax²-5x + c are equal to zero each,find the values of a and c

Solution

Given Polynomial,

$\sf \: p(x) = a {x}^{2} - 5x + c$

• Sum of Zeros = 10

• Product of Zeros = 10

Note

• $\sf Sum \ Of \ Zeros = - \dfrac{x \ coefficient }{x^2 \ coefficient}$
• $\sf {Product \ Of \ Zeros = \dfrac{Constant \ Term}{x^2 \ coefficient }}$

According to the Question,

• For the value of a

$\sf \: - ( - \dfrac{5}{a}) = 10 \\ \\ \longrightarrow \: \sf \: a = \dfrac{5}{10} \\ \\ \longrightarrow \: \boxed{\boxed{ \sf \: a = \dfrac{1}{2} }}$

• For the value of c

$\sf \: \dfrac{c}{a} = 10 \\ \\ \longrightarrow \sf \: 2c = 10 \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: c = 5}}$

Answer 2

$\underline{ \boxed{ \mathfrak{ \huge{ \pink{Answer}}}}} \\ \\ \star \rm \: \orange{Given} \\ \\ \implies \rm \: p(x) = a {x}^{2} - 5x + c \\ \\ \implies \rm \: sum \: of \: zeroes = 10 \\ \\ \implies \rm \: product \: of \: zeroes = 10 \\ \\ \star \rm \: \orange{To \: Find} \\ \\ \implies \rm \: value \: of \: a \: and \: c..... \\ \\ \star \rm \: \orange{Formula} \\ \\ \implies \rm \: polynomial \: {eq}^{n} \: \boxed{ \rm{\red{p(x) = a {x}^{2} + bx + c}}} \\ \\ \implies \rm \: sum \: of \: zeroes = \boxed{ \rm\blue{\frac{ - b}{a}}} \\ \\ \implies \rm \: product \: of \: zeroes = \boxed{ \rm{ \green{\frac{c}{a} }}} \\ \\ \star \rm \: \orange{Calculation} \\ \\ \implies \rm \: comparing \: given \: {eq}^{n} \: with \: polynomial \: {eq}^{n} \\ \\ \rm \: we \: get \: b = - 5 \\ \\ \leadsto \rm \: sum \: of \: zeroes = \frac{ - b}{a} \\ \\ \therefore \rm \: 10 = \frac{ - ( - 5)}{a} \\ \\ \therefore \rm \: \boxed{ \red{ \rm{a = \frac{1}{2} }}} \: \purple{\star} \\ \\ \leadsto \rm \: product \: of \: zeroes = \frac{c}{a} \\ \\ \therefore \rm \: 10 = \frac{c}{ \frac{1}{2} } \\ \\ \therefore \: \boxed{ \rm{ \red{c = 5}}}\: \purple{\star}$