If x⁴ + 2x³ + ax² + bx + 9 is a perfect square where a and b are positive real numbers, then the value of a and b is?

1a = 5, b = 6
2a = 6, b = 7
3a = 7, b = 7
4a = 7, b = 8

Answer:- 1

Explanation:-

Solution:

\begin{aligned} x^{4} + 2 x^{3} + a x^{2} + b x + 9 \\ Put x = 1 \\ = 1 + 2 \times 1 + a + b + 9 \\ = 1 + 2 + a + b + 9 \\ = 13 + a + b \end{aligned}

$\eqalign{ & {x^4} + 2{x^3} + a{x^2} + bx + 9 \cr & {\text{Put }}x = 1 \cr & = 1 + 2 \times 1 + a + b + 9 \cr & = 1 + 2 + a + b + 9 \cr & = 13 + a + b \cr}$
To make a perfect square numbers value of a + b must be either 3 or 13
Now, option (2) a = 6, b = 7

\begin{aligned} ∴ a + b = 12 \\ make perfect square \\ (25 = 5^{2}) \end{aligned}

$\eqalign{ & \therefore a + b = 12 \cr & {\text{make perfect square}} \cr & \left( {25 = {5^2}} \right) \cr}$

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To make a perfect square numbers value of a + b must be either 3 or 13
Now, option (2) a = 6, b = 7

\begin{aligned} ∴ a + b = 12 \\ make perfect square \\ (25 = 5^{2}) \end{aligned}

$\eqalign{ & \therefore a + b = 12 \cr & {\text{make perfect square}} \cr & \left( {25 = {5^2}} \right) \cr}$ ", "dateCreated": "7/24/2019 10:09:12 AM", "author": { "@type": "Person", "name": "Nitin Sir" } }, "suggestedAnswer": { "@type": "Answer", "text": "

Solution:

\begin{aligned} x^{4} + 2 x^{3} + a x^{2} + b x + 9 \\ Put x = 1 \\ = 1 + 2 \times 1 + a + b + 9 \\ = 1 + 2 + a + b + 9 \\ = 13 + a + b \end{aligned}

\begin{aligned} ∴ a + b = 12 \\ make perfect square \\ (25 = 5^{2}) \end{aligned}

$\eqalign{ & \therefore a + b = 12 \cr & {\text{make perfect square}} \cr & \left( {25 = {5^2}} \right) \cr}$ ", "dateCreated": "7/24/2019 10:09:12 AM" } }