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Lapatulllka [165]
2 years ago
6

Find the value of c that completes the square

Mathematics
1 answer:
PilotLPTM [1.2K]2 years ago
6 0

Answer:

The value of c that completes the square  x^2+2x+c is 1

Step-by-step explanation:

We need to find the value of c that completes the square  x^2+2x+c

The formula used will be: a^2+2ab+b^2=(a+b)^2

In the question given the value c can be found by breaking the middle term. we are given 2x while the general formula is 2ab for middle term so,

2(x)(1) = 2x

so, c= 1

Solving:

x^2+2x+c\\=x^2+2(x)(1)+(1)^2-(1)^2\\=(x+1)^2-1

So, the value of c that completes the square  x^2+2x+c is 1

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For the composite function, identify an inside function and an outside function and write the derivative with respect to x of th
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Answer:

The inner function is h(x)=4x^2 + 8 and the outer function is g(x)=3x^5.

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Step-by-step explanation:

A composite function can be written as g(h(x)), where h and g are basic functions.

For the function f(x)=3(4x^2+8)^5.

The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function.

Here, we have 4x^2+8 inside parentheses. So h(x)=4x^2 + 8 is the inner function and the outer function is g(x)=3x^5.

The chain rule says:

\frac{d}{dx}[f(g(x))]=f'(g(x))g'(x)

It tells us how to differentiate composite functions.

The function f(x)=3(4x^2+8)^5 is the composition, g(h(x)), of

     outside function: g(x)=3x^5

     inside function: h(x)=4x^2 + 8

The derivative of this is computed as

\frac{d}{dx}\left(3\left(4x^2+8\right)^5\right)=3\frac{d}{dx}\left(\left(4x^2+8\right)^5\right)\\\\\mathrm{Apply\:the\:chain\:rule}:\quad \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}\\f=u^5,\:\:u=\left(4x^2+8\right)\\\\3\frac{d}{du}\left(u^5\right)\frac{d}{dx}\left(4x^2+8\right)\\\\3\cdot \:5\left(4x^2+8\right)^4\cdot \:8x\\\\120x\left(4x^2+8\right)^4

The derivative of the function is \frac{d}{dx}\left(3\left(4x^2+8\right)^5\right)=120x\left(4x^2+8\right)^4.

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