Question

what must be added to 9 x -24 x+16 to make it perfect square​

Answer 1

First I thought that the polynomial must be 9x² - 24x + 16.

$\Longrightarrow\ 9x^2-24x+16 \\ \\ \Longrightarrow\ (3x)^2-(2 \times 3x \times 4)+(4^2) \\ \\ \Longrightarrow\ (3x-4)^2$

Here it seems that the polynomial is itself a perfect square. So there's no need to add it to make it another perfect square.

Thus the answer would be 0.

But according to 9x - 24x + 16 which is mentioned in the question,

$\displaystyle \Longrightarrow\ 9x-24x+16 \\ \\ \\ \Longrightarrow\ -15x+16 \\ \\ \\ \Longrightarrow\ 16-15x \\ \\ \\ \Longrightarrow\ (4)^2-\left(2 \times 4 \times \frac{15}{8}x\right)+\left(\frac{15}{8}x\right)^2-\left(\frac{15}{8}x\right)^2 \\ \\ \\ \Longrightarrow\ \left(4-\frac{15}{8}x\right)^2-\ \frac{225}{64}x^2$

Here, it seems that the polynomial 9x - 24x + 15 is 225x²/64, i.e., (15x/8)² is subtracted from a perfect square. From this, we can find that when we add 225x²/64 to this polynomial, it becomes a perfect square.

Thus the answer would be 225x²/64.

Answer 2

Answer:

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