Question

Help with solving for x in the equation (2x+3)^2 +x^2 = 40000

Answer 1

first u can try simplify this

\(\large \begin{align} \color{black}{(2x+3)^2 =\hspace{.33em}\\~\\}\end{align}\)

Answer 2

4x^2 +12x +9

Answer 3

\(\large \begin{align} \color{black}{(2x+3)^2 +x^2=40000\hspace{.33em}\\~\\
4x^2 +12x +9 +x^2=40000\hspace{.33em}\\~\\
5x^2 +12x +9 =40000\hspace{.33em}\\~\\
x^2 +\dfrac{12x}{5} =\dfrac{39991}{5}\hspace{.33em}\\~\\
x^2 +2\cdot \dfrac{6x}{5} =\dfrac{39991}{5}\hspace{.33em}\\~\\
x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2 =\dfrac{39991}{5}+ (\dfrac{6}{5})^2 \hspace{.33em}\\~\\
}\end{align}\)

Answer 4

\(\large \begin{align} \color{black}{

x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2 =\dfrac{39991}{5}+ (\dfrac{6}{5})^2 \hspace{.33em}\\~\\
(x+\dfrac{6}{5})^2 =\dfrac{39991}{5}+ \dfrac{36}{25} \hspace{.33em}\\~\\
}\end{align}\)

Answer 5

Where does dividing by 5 come from?

Answer 6

i divided this equation by 5

\(\large \begin{align} \color{black}{

5x^2 +12x +9 =40000\hspace{.33em}\\~\\ \hspace{.33em}\\~\\
}\end{align}\)

Answer 7

do u know squaring method

Answer 8

Is it the method used to solve (equation) = 0 equations?

Answer 9

yes

Answer 10

i wanted to make the red part a perfect squarw

\(\large \begin{align} \color{black}{

\color{red}{x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2} =\dfrac{39991}{5}+ (\dfrac{6}{5})^2 \hspace{.33em}\\~\\
}\end{align}\)

Answer 11

Ok

Answer 12

In the first step with dividing by 5, you are getting rid of 5 by dividing both sides of the equation by 5, right?

Answer 13

Then reducing the 12x to 6x by moving a multiplication of 2 to the left side of the fraction(changing the form but not the value)

Answer 14

yes i want to get rid of 5 from the \(x^2\) term

Answer 15

yes u r correct!

Answer 16

What about (6/5)^2 on both sides of the equation?

Answer 17

i applied this to red part

\(\large \begin{align} \color{black}{
a^2+2ab+b^2=(a+b)^2 \hspace{.33em}\\~\\

}\end{align}\)

Answer 18

\(\large \begin{align} \color{black}{

x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2 =\dfrac{39991}{5}+ (\dfrac{6}{5})^2 \hspace{.33em}\\~\\
(x+\dfrac{6}{5})^2 =\dfrac{199991}{25} \hspace{.33em}\\~\\
(x+\dfrac{6}{5}) =\pm\sqrt{\dfrac{199991}{25}} \hspace{.33em}\\~\\
(x+\dfrac{6}{5}) =\pm\dfrac{1}{5}\sqrt{199991} \hspace{.33em}\\~\\

}\end{align}\)

Answer 19

So the format of this equation would take b but not a(removing the fraction but not x) and add it on both sides? Also, how is the left side reduced to \[(x+6/5)^2\]

Answer 20

\(\large \begin{align} \color{red}{
x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2 \\
=x^2+ \dfrac{6x}{5}+\dfrac{6x}{5}+ (\dfrac{6}{5})^2 \\
=x(x+ \dfrac{6}{5})+\dfrac{6}{5}(x+ (\dfrac{6}{5})) \\
=(x+ \dfrac{6}{5})(x+ (\dfrac{6}{5})) \\
=(x+ \dfrac{6}{5})^2\\

}\end{align}\)


see if that makes sense

Answer 21

So the multiplication of 2 makes 6x/5 become 2 fractions. The x is distributed in both fractions to change the form of everything but what happens to the 6/5 in the middle and the x on the outside?

Answer 22

\(\large \begin{align} \color{red}{
x^2 +2\cdot \dfrac{6x}{5}+ (\dfrac{6}{5})^2 \\
=x^2+ \dfrac{6x}{5}+\dfrac{6x}{5}+ (\dfrac{6}{5})^2 \\
=\color{blue}{x}(x+ \dfrac{6}{5})+\color{blue}{\dfrac{6}{5}}(x+ (\dfrac{6}{5})) \\
=(x+ \dfrac{6}{5})(x+ (\dfrac{6}{5})) \\
=(x+ \dfrac{6}{5})^2\\

}\end{align}\)

i just took out x and \(6x/5\) common

Answer 23

Ok i understand now

Answer 24

i mean x and 6/5 not 6x/5

Answer 25

u know how to proceed from here

\((x+\dfrac{6}{5}) =\pm\dfrac{1}{5}\sqrt{199991} \hspace{.33em}\\~\\\)

Answer 26

\(\large \begin{align} \color{black} {

(x+\dfrac{6}{5}) =\pm\dfrac{1}{5}\sqrt{199991}
\hspace{.33em}\\~\\
(x+\dfrac{6}{5}) =+\dfrac{1}{5}\sqrt{199991} \hspace{.33em}\\~\\
\hspace{.33em}\\~\\
or \hspace{.33em}\\~\\
(x+\dfrac{6}{5}) =-\dfrac{1}{5}\sqrt{199991}\hspace{.33em}\\~\\
so \hspace{.33em}\\~\\
x =-\dfrac{1}{5}\sqrt{199991}-\dfrac{6}{5} \hspace{.33em}\\~\\

x =+\dfrac{1}{5}\sqrt{199991}-\dfrac{6}{5} \hspace{.33em}\\~\\
}\end{align}\)

Answer 27

check with wolfram

http://www.wolframalpha.com/input/?i=solve+%282x%2B3%29%5E2+%2Bx%5E2+%3D+40000

Answer 28

Ok, i understand it now. Thanks for the help.