How do you solve $4(x+1)^2=100$ ?

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Answer 1 · 2016-08-01T12:35:59

$x = 4" "$ or $" "x = -6$

The key to this problem is the fact that for a given number $a in RR$ , you have

$a^2 = a * a" "$ and $" "a^2 = (-a) * (-a)$

so right from the start you know that you're looking for two values of $x$ , since for $a in RR$ , you have

$color(purple)(|bar(ul(color(white)(a/a)color(black)(sqrt(a^2) = |a| = {(+ a, " for "a >=0), (-a, " for "a < 0):})color(white)(a/a)|)))$

The first thing to do here is divide both sides of the equation by $4$

$(color(red)(cancel(color(black)(4))) * (x+1)^2)/color(red)(cancel(color(black)(4))) = 100/4$

$(x+1)^2 = 25$

If you take the square root of both sides of the equation

$sqrt((x+1)^2) = sqrt(25)$

you will end up with

$(x+1) = + 5" "$ or $" "(x+1) = -5$

This will get you

$x+1 = 5 implies x= 4" "$ or $" " x+1 = -5 implies x= -6$

Therefore, the original equation has two possible solutions

$x = 4" "$ or $" "x = -6$

Do a quick check to make sure that the calculations are correct

$x = 4:" " 4 * (4 + 1)^2 = 100$

$4 * 5^2 = 100" "color(green)(sqrt())$

$x = -6:" " 4 * (-6 + 1)^2 = 100$

$4 * (-5)^2 = 100" "color(green)(sqrt())$

How do you solve 4(x+1)^2=1004(x+1)^2=100?