How do you solve $x^{2} + 10 x + 5 = 0$ $x^2 + 10x + 5 = 0$ by completing the square?

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How do you solve $x^{2} + 10 x + 5 = 0$ $x^2 + 10x + 5 = 0$ by completing the square?

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Answer 1 · 2016-05-13T12:01:21

$x = - 5 \pm 2 \sqrt{5}$ $x=-5+-2sqrt(5)$
(see below for completing the squares method of solution)

Explanation:

Given:
$XXX x^{2} + 10 x + 5 = 0$ $color(white)("XXX")x^2+10x+5=0$

Move the constant to the right side as
$XXX x^{2} + 10 x = - 5$ $color(white)("XXX")color(blue)(x^2+10x)=-5$

We know that ${(x + a)}^{2} = x^{2} + 2 a x + a^{2}$ $(x+a)^2=color(red)(x^2+2ax+a^2)$
So if the first two terms of a squared binomial are
$XXX x^{2} + 2 a x = x^{2} + 10 x$ $color(white)("XXX")color(red)(x^2+2ax) = color(blue)(x^2+10x)$
then
$XXX a = 5$ $color(white)("XXX")color(red)(a)=color(blue)(5)$
and we will need to add
$XXX a^{2} = 25$ $color(white)("XXX")color(red)(a^2)=color(blue)(25)$ (to both sides) to complete the square:

$XXX x^{2} + 10 x + 25 = - 5 + 25$ $color(white)("XXX")x^2+10x+25 = -5+25$

Writing as a squared binomial and simplifying the right side:
$XXX {(x + 5)}^{2} = 20$ $color(white)("XXX")(x+5)^2=20$

Taking the square root of both sides:
$XXX x + 5 = \pm 2 \sqrt{5}$ $color(white)("XXX")x+5=+-2sqrt(5)$

Subtracting $5$ $5$ from both sides
$XXX x = - 5 \pm 2 \sqrt{5}$ $color(white)("XXX")x=-5+-2sqrt(5)$

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How do you solve $x^{2} + 10 x + 5 = 0$ $x^2 + 10x + 5 = 0$ by completing the square?

1 Answer

Explanation:

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