Question #451c2

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Question #451c2

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Answer 1 · 2017-03-01T21:52:12

$x = \frac{3}{2} \pm \frac{\sqrt{21}}{2} \leftarrow exact values$ $x=3/2+-sqrt(21)/2 larr" exact values"$

$x \approx 3.791; x \approx - 0.791 \leftarrow approximate values$ $x~~3.791" ; "x~~-0.791 larr" approximate values"$

Explanation:

$This part in great detail to demonstrate first principles$ $color(brown)("This part in great detail to demonstrate first principles")$

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

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

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

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

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

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

Subtract $3 x$ $color(red)(3x)$ from both sides

$3 x - 3 x = x^{2} - 3 - 3 x$ $color(green)(" "3xcolor(red)(-3x)" "=" "x^2-3color(red)(-3x)$

$0 = x^{2} - 3 x - 3$ $" "0" "=" "x^2-3x-3$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Calculated much faster now$ $color(brown)("Calculated much faster now")$

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

where $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x=(-b+-sqrt(b^2-4ac))/(2a)$

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

$x = \frac{3}{2} \pm \frac{\sqrt{9 + 12}}{2}$ $x=3/2+-sqrt(9+12)/2$

$x = \frac{3}{2} \pm \frac{\sqrt{21}}{2} \leftarrow exact values$ $x=3/2+-sqrt(21)/2 larr" exact values"$

$x \approx 3.791; x \approx - 0.791$ $x~~3.791" ; "x~~-0.791$

Tony B

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Question #451c2

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