How do you solve 0 = x^2 + 10x + 5?

2 Answers
George C.
May 12, 2016

x = -5+-2sqrt(5)

Explanation:

Complete the square and use the difference of squares identity:

a^2-b^2=(a-b)(a+b)

with a=(x+5) and b=2sqrt(5) as follows:

0 = x^2+10x+5

=(x+5)^2-25+5

=(x+5)^2-20

=(x+5)^2-(2sqrt(5))^2

=((x+5)-2sqrt(5))((x+5)+2sqrt(5))

=(x+5-2sqrt(5))(x+5+2sqrt(5))

Hence:

x = -5+-2sqrt(5)

Alan P.
May 12, 2016

x=-5+2sqrt(5)color(white)("XXX")orcolor(white)("XXX")x=-5-2sqrt(5)

Explanation:

The easiest way for this particular example is to use the quadratic formula:

For an equation of the form color(red)(a)x^2+color(blue)(b)x+color(green)(c)=0
the solutions are given by x=(-color(blue)(b)+-sqrt(color(blue)(b)^2-4color(red)(a)color(green)(c)))/(2color(red)(a))

For the given example

color(red)(a)=1
color(blue)b=10 and
color(green)c=5

So
color(white)("XXX")x=(-10+-sqrt(10^2-4(1)(5)))/(2(1))

color(white)("XXXX")=(-10+-4sqrt(5))/2

color(white)("XXXX")=-5+-2sqrt(5)

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