# Question 451c2

Mar 1, 2017

$x = \frac{3}{2} \pm \frac{\sqrt{21}}{2} \leftarrow \text{ exact values}$

$x \approx 3.791 \text{ ; "x~~-0.791 larr" approximate values}$

#### Explanation:

$\textcolor{b r o w n}{\text{This part in great detail to demonstrate first principles}}$

Multiply everything inside the bracket by the $x$ that is outside giving:

$\text{ "3x-x^2" "=" } - 3$

To make the ${x}^{2}$ positive add ${x}^{2}$ to both sides

" "color(green)(3x-x^2color(red)(+x^2)" "=" "-3color(red)(+x^2)) 

But $- {x}^{2} + {x}^{2} = 0$

$\text{ "3x+0" "=" } {x}^{2} - 3$

Subtract $\textcolor{red}{3 x}$ from both sides

color(green)(" "3xcolor(red)(-3x)" "=" "x^2-3color(red)(-3x)#

$\text{ "0" "=" } {x}^{2} - 3 x - 3$
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$\textcolor{b r o w n}{\text{Calculated much faster now}}$

Standard form $\to y = a {x}^{2} + b x + c$

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

$x = \frac{- \left(- 3\right) \pm \sqrt{{\left(- 3\right)}^{2} - 4 \left(1\right) \left(- 3\right)}}{2 \left(1\right)}$

$x = \frac{3}{2} \pm \frac{\sqrt{9 + 12}}{2}$

$x = \frac{3}{2} \pm \frac{\sqrt{21}}{2} \leftarrow \text{ exact values}$

$x \approx 3.791 \text{ ; } x \approx - 0.791$