# The length of a rectangle plus its width is 25 cm. The area is 156 square cm. What are the length and width of the rectangle?

Aug 5, 2016

Length$= 13$
Width$= 12$

#### Explanation:

Let Length of the rectangle be $x$
Let Width of the rectangle be $y$
so
$x + y = 25$
Area $= x y = 156$
or
$2 x y = 312$
${\left(x + y\right)}^{2} = {25}^{2} = 625$
or
${x}^{2} + {y}^{2} + 2 x y = 625$
or
${x}^{2} + {y}^{2} + 312 = 625$
or
${x}^{2} + {y}^{2} = 313$
Therefore
${\left(x - y\right)}^{2} = {x}^{2} + {y}^{2} - 2 x y$
or
${\left(x - y\right)}^{2} = 313 - 312$
or
${\left(x - y\right)}^{2} = 1$
or
$x - y = \sqrt{1}$
or
$x - y = 1$
and we have been given
$x + y = 25$
we get
$x + x - y + y = 25 + 1$
or
$2 x + 0 = 26$
or
$2 x = 26$
or
$x = \frac{26}{2}$
or
$x = 13$-------------Ans $1$
By putting the value of $x = 13$ in the equation $x + y = 25$
We get
$13 + y = 25$
or
$y = 25 - 13$
or
$y = 12$--------------Ans $2$

Aug 5, 2016

$w = + 12 \mathmr{and} + 13$

#### Explanation:

Let length be $L$
Let width be $w$
Lat area be $a$

Then $L + w = 25$

$\implies L = 25 - w$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$a = 156 = L w = w \left(25 - w\right)$

$\implies - {w}^{2} + 25 w - 156 = 0$

Multiply everything by (-1)

$\implies {w}^{2} - 25 w + 156 = 0$

Not that $\left(- 12\right) \times \left(- 13\right) = + 156$

And that $- 12 - 13 = - 25$

$\left(w - 12\right) \left(w - 13\right) = 0$

$w = + 12 \mathmr{and} + 13$