How do you solve $f (x) = 2 x^{2} - 12 x + 17$ $f(x)=2x^2-12x+17$ by completing the square?

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How do you solve $f (x) = 2 x^{2} - 12 x + 17$ $f(x)=2x^2-12x+17$ by completing the square?

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Answer 1 · 2017-04-23T08:03:13

$3 \pm \sqrt{\frac{1}{2}}$ $3 +- sqrt(1/2)$

Explanation:

$f (x) = 2 x^{2} - 12 x + 17 \Rightarrow 2 (x^{2} - 6 x) + 17$ $f(x) = 2x^2 - 12x + 17 => 2(x^2 - 6x) + 17$

so,

$2 {(x - 3)}^{2} - 18 + 17$ $2(x - 3)^2 - 18 + 17$

Hence,

$2 x^{2} - 12 x + 17 = 2 {(x - 3)}^{2} - 1$ $2x^2 - 12x + 17 = 2(x - 3)^2 - 1$

By 'solving' I assume you mean $2 x^{2} - 12 x + 17 = 0$ $2x^2 - 12x + 17 = 0$

i.e. $2 {(x - 3)}^{2} - 1 = 0$ $2(x - 3)^2 - 1 = 0$

$\Rightarrow {(x - 3)}^{2} = \frac{1}{2}$ $=> (x - 3)^2 = 1/2$

$\Rightarrow x - 3 = \pm \sqrt{\frac{1}{2}}$ $=> x - 3 = ± sqrt(1/2)$

so, $x = 3 \pm \sqrt{\frac{1}{2}}$ $x = 3 ± sqrt(1/2)$

:)>

Answer 2 · 2017-04-23T08:26:50

$x = 3 \pm \frac{\sqrt{2}}{2}$ $x=3+-sqrt2/2$

Explanation:

To $complete the square$ $color(blue)"complete the square"$

add ${(\frac{1}{2} coefficient of x-term)}^{2}$ $(1/2" coefficient of x-term")^2$

Require coefficient of $x^{2}$ $x^2$ term to be 1

$f (x) = 2 (x^{2} - 6 x) + 17$ $f(x)=2(x^2-6x)+17$

$f (x) = 2 (x^{2} - 6 x + 9 - 9) + 17$ $color(white)(f(x))=2(x^2-6xcolor(red)(+9 -9))+17$

Since we have added +9 which is not there we must also subtract 9

$f (x) = 2 {(x - 3)}^{2} - 18 + 17$ $f(x)=2(x-3)^2-18+17$

$\Rightarrow f (x) = 2 {(x - 3)}^{2} - 1$ $rArrf(x)=2(x-3)^2-1$

To solve $equate f(x) to zero$ $color(blue)"equate f(x) to zero"$

$\Rightarrow 2 {(x - 3)}^{2} - 1 = 0$ $rArr2(x-3)^2-1=0$

$\Rightarrow 2 {(x - 3)}^{2} = 1$ $rArr2(x-3)^2=1$

$\Rightarrow {(x - 3)}^{2} = \frac{1}{2}$ $rArr(x-3)^2=1/2$

$take the square root of both sides$ $color(blue)"take the square root of both sides"$

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

$\Rightarrow x - 3 = \pm \frac{1}{\sqrt{2}}$ $rArrx-3=+-1/sqrt2$

$\Rightarrow x = 3 \pm \frac{1}{\sqrt{2}} = 3 \pm \frac{\sqrt{2}}{2} \leftarrow rationalise denominator$ $rArrx=3+-1/sqrt2=3+-sqrt2/2larrcolor(red)" rationalise denominator"$

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How do you solve $f (x) = 2 x^{2} - 12 x + 17$ $f(x)=2x^2-12x+17$ by completing the square?

2 Answers

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