New user? Sign up

Existing user? Log in

If \(x + y \ = \ 3\) and \(x^{3} + y^{3} \ = \ 25\), what is \(x^{2} + y^{2}\)

Note by Paulo Carlos 3 years ago

Easy Math Editor

*italics*

_italics_

**bold**

__bold__

- bulleted- list

1. numbered2. list

paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

> This is a quote

This is a quote

# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

2 \times 3

2^{34}

a_{i-1}

\frac{2}{3}

\sqrt{2}

\sum_{i=1}^3

\sin \theta

\boxed{123}

Sort by:

Cubing the first equation, \(x^3+3x^2y+3xy^2+y^3=27=>3xy(x+y)=27-(x^3+y^3)=2\), so \(xy=\frac{2}{9}\). Now, note that \(x^2+y^2=(x+y)^2-2xy=3^2-2\times\frac{2}{9}=\boxed{\frac{77}{9}}\).

Log in to reply

Just for the sake of mentioning, one can overkill this problem using Newton's Identities:

\[x^2+y^2=(x+y)^2-2xy\implies x^2+y^2=9-2xy\]

\[x^3+y^3=(x+y)(x^2+y^2)-xy(x+y)\implies 25=3(x^2+y^2-xy)\implies 25=3(9-2xy-xy)\implies xy=\frac 29\]

We resubstitute this value of \(xy\) back to our first equation to get \(x^2+y^2=\dfrac{77}{9}\)

I think you can perhaps use \(\rightarrow\) OR \(\Rightarrow\) instead of => .

The \(\LaTeX\) codes are \rightarrow and \Rightarrow respectively .

\rightarrow

\Rightarrow

\[x^3+y^3 = (x+y)(x^2-xy+y^2) \\\Rightarrow 25 = 3(x^2-xy+y^2) \\\Rightarrow x^2-xy+y^2=\dfrac{25}{3} \dots (1) \\\Rightarrow (x+y)^2=3^2 \\\Rightarrow x^2+2xy+y^2=9 \\ \Rightarrow \dfrac{x^2}{2} + xy + \dfrac{y^2}{2} = \dfrac{9}{2} \dots (2) \]

Adding \((1),(2)\) , we have:

\[\dfrac{3x^2}{2}+\dfrac{3y^2}{2}= \dfrac{25}{3}+\dfrac{9}{2} \\\Rightarrow \dfrac{3}{2}(x^2+y^2)= \dfrac{77}{6} \\ \Rightarrow \boxed{x^2+y^2= \dfrac{77}{9}} \]

hm....What is the answer?

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestCubing the first equation, \(x^3+3x^2y+3xy^2+y^3=27=>3xy(x+y)=27-(x^3+y^3)=2\), so \(xy=\frac{2}{9}\). Now, note that \(x^2+y^2=(x+y)^2-2xy=3^2-2\times\frac{2}{9}=\boxed{\frac{77}{9}}\).

Log in to reply

Just for the sake of mentioning, one can overkill this problem using Newton's Identities:

\[x^2+y^2=(x+y)^2-2xy\implies x^2+y^2=9-2xy\]

\[x^3+y^3=(x+y)(x^2+y^2)-xy(x+y)\implies 25=3(x^2+y^2-xy)\implies 25=3(9-2xy-xy)\implies xy=\frac 29\]

We resubstitute this value of \(xy\) back to our first equation to get \(x^2+y^2=\dfrac{77}{9}\)

Log in to reply

I think you can perhaps use \(\rightarrow\) OR \(\Rightarrow\) instead of => .

The \(\LaTeX\) codes are

`\rightarrow`

and`\Rightarrow`

respectively .Log in to reply

\[x^3+y^3 = (x+y)(x^2-xy+y^2) \\\Rightarrow 25 = 3(x^2-xy+y^2) \\\Rightarrow x^2-xy+y^2=\dfrac{25}{3} \dots (1) \\\Rightarrow (x+y)^2=3^2 \\\Rightarrow x^2+2xy+y^2=9 \\ \Rightarrow \dfrac{x^2}{2} + xy + \dfrac{y^2}{2} = \dfrac{9}{2} \dots (2) \]

Adding \((1),(2)\) , we have:

\[\dfrac{3x^2}{2}+\dfrac{3y^2}{2}= \dfrac{25}{3}+\dfrac{9}{2} \\\Rightarrow \dfrac{3}{2}(x^2+y^2)= \dfrac{77}{6} \\ \Rightarrow \boxed{x^2+y^2= \dfrac{77}{9}} \]

Log in to reply

hm....What is the answer?

Log in to reply