If x^3 + 1/x^3 = 110 then find x + 1/x

1 answer:

3 0

$Use:(a+b)^3=a^3+3a^2b+3ab^2+b^3\ (*)\\\\x^3+\left(\dfrac{1}{x}\right)^3\\=\underbrace{x^3+3\cdot x^2\cdot\dfrac{1}{x}+3\cdot x\cdot\left(\dfrac{1}{x}\right)^2+\left(\dfrac{1}{x}\right)^3}_{(*)}-3\cdot x^2\cdot\dfrac{1}{x}-3\cdot x\cdot\left(\dfrac{1}{x}\right)^2\\\\=\left(x+\dfrac{1}{x}\right)^3-3x-3\cdot\dfrac{1}{x}=\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)$

therefore

$x^3+\left(\dfrac{1}{x}\right)^3=110\iff \left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)=110\\\\subtitute\ t=x+\dfrac{1}{x}\\\\t^3-3t=110\ \ \ \ |subtract\ 110\ from\ both\ sides\\\\t^3-3t-110=0\\\\t^3-25t+22t-110=0\\\\t(t^2-25)+22(t-5)=0\ \ \ |use\ a^2-b^2=(a-b)(a+b)\\\\t(t-5)(t+5)+22(t-5)=0\\\\(t-5)[t(t+5)+22]=0\iff t-5=0\ or\ \underbrace{t^2+5t+22=0}_{no\ solution}\\\\t=5\iff x+\dfrac{1}{x}=5\\\\Answer:\boxed{x+\dfrac{1}{x}=5}$

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Why does (-4 1/2)^2 = 20.25? Thank you please hurry!

AnnZ [28]

Answer:

See below.

Step-by-step explanation:

Squaring a number means to multiply the number by itself.

For example, 3^2 means 3 * 3.

8^2 means 8 * 8.

(-6)^2 means (-6) * (-6).

In each case above, you see that the number that is squared is multiplied by itself.

In this case, (-4.5)^2, squaring the number -4.5, means multiply the number -4.5 by the number -4.5

(-4.5)^2 = (-4.5) * (-4.5)

Now we need to actually multiply -4.5 by -4.5

When you multiply a negative number by a negative number, the result is positive, so the product of -4.5 and -4.5 is a positive number. To find that number, you multiply 4.5 by 4.5

4.5 * 4.5 = 20.25

Therefore, (-4.5)^2 = 20.25

4 0

3 years ago

Muriel says she has written a system of two linear equations that has an infinite number of solutions. One of the equations of t

JulijaS [17]

A system of equations with infinitely many solutions is a system where the two equations are identical. The lines coincide. Anything that is equal to $3y=2x-9$ will work. You could try multiply the entire equation by some number, or moving terms around, or adding terms to both sides, or any combination of operations that you apply to the entire equation.

You could multiply the whole thing by 4.5 to get $13.5y=9x-40.5$ . If you want, you could mix things up and write it in slope-intercept form: $y= \frac{2}{3}x-3$ . The point is, anything that is equivalent to the original equation will give infinitely many solutions x and y. You can test this by plugging in values x and y and seeing the answers!

The attached graph shows that four different equations are really the same.