How do you solve sqrt(x+6)=x?

1 Answer
Stefan V.
Oct 12, 2015

x=3

Explanation:

Start by squaring both sides of the equation to get rid of the square root

(sqrt(x+6))^2 = x^2

x + 6 = x^2

Next, move all the tems on one side of the equation

x^2 - x - 6 = 0

Use the quadratic formula to find the two roots of this quadratic equation

x_(1, 2) = (-(-1) +- sqrt( (-1)^2 - 4 * 1 * (-6)))/(2 * 1)

x_(1,2) = (1 +- sqrt(25))/2 = (1 +- 5)/2 = { (x_1 = (1 + 5)/2 = 3), (x_2 = (1 - 5)/2 = -2) :}

Notice that x=-2 does not satisfy the initial equation, since you would get

sqrt(-2 + 6) = -2

color(red)(cancel(color(black)(sqrt(4) = -2)))

This means that x=-2 will eb an extraneous solution. The only valid solution will thus be x = 3, for which

sqrt(3 + 6) = 3

sqrt(9) = 3 " "color(green)(sqrt())

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