# How do you solve x^2-4.7x=-2.8 by completing the square?

Apr 23, 2017

color(red)(x=4 or color(red)(x=0.7

#### Explanation:

$\therefore {x}^{2} - 4.7 x + 2.8 = 0$

$\therefore {x}^{2} - 4.7 x + {\left(- \frac{4.7}{2}\right)}^{2} = - 2.8 + {\left(- \frac{4.7}{2}\right)}^{2}$

$\therefore {x}^{2} - 4.7 x + 5.5225 = - 2.8 + 5.5225$

$\therefore {\left(x - 2.35\right)}^{2} = 2.7225$

$\therefore \sqrt{{\left(x - 2.35\right)}^{2}} = \sqrt{2.7225}$

$\therefore x - 2.35 = \pm 1.65$

$\therefore x = 2.35 \pm 1.65$

:.color(red)(x=2.35+1.65=4

or :.color(red)(x=2.35-1.65=0.7

check:

$\therefore x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$\therefore x = \frac{- \left(- 4.7\right) \pm \sqrt{{\left(- 4.7\right)}^{2} - 4 \left(1\right) \left(2.8\right)}}{2 a}$

$\therefore x = \frac{- \left(- 4.7\right) \pm \sqrt{22.09 - 11.2}}{2 a}$

$\therefore x = \frac{4.7 \pm \sqrt{10.89}}{2}$

$\therefore x = \frac{4.7 \pm 3.3}{2}$

$\therefore x = \frac{4.7 + 3.3}{2}$

:.color(red)(x=8/2=4

or $\therefore x = \frac{4.7 - 3.3}{2}$

:.color(red)(x=1.4/2=0.7

Apr 23, 2017

$x = 4 \text{ }$ or $\text{ } x = 0.7$

#### Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Use this with $a = \left(20 x - 47\right)$ and $b = 33$ below.

Given:

${x}^{2} - 4.7 x = - 2.8$

Add $2.8$ to both sides to get:

${x}^{2} - 4.7 x + 2.8 = 0$

Multiply through by $400$ to allow us to complete the square using integers...

$0 = 400 \left({x}^{2} - 4.7 x + 2.8\right)$

$\textcolor{w h i t e}{0} = 400 {x}^{2} - 1880 x + 1120$

$\textcolor{w h i t e}{0} = {\left(20 x\right)}^{2} - 2 \left(20 x\right) \left(47\right) + {47}^{2} - 1089$

$\textcolor{w h i t e}{0} = {\left(20 x - 47\right)}^{2} - {33}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(20 x - 47\right) - 33\right) \left(\left(20 x - 47\right) + 33\right)$

$\textcolor{w h i t e}{0} = \left(20 x - 80\right) \left(20 x - 14\right)$

$\textcolor{w h i t e}{0} = 40 \left(x - 4\right) \left(10 x - 7\right)$

So:

$x = 4 \text{ }$ or $\text{ } x = \frac{7}{10} = 0.7$