First, subtract color(red)(10x) from each side of the equation to isolate the fraction while keeping the equation balanced:
[(x^2 + 5x)/(x + 2)] + 10x - color(red)(10x) = -24 - color(red)(10x)
[(x^2 + 5x)/(x + 2)] + 0 = -24 - 10x
[(x^2 + 5x)/(x + 2)] = -10x - 24
Next, multiply each side of the equation by color(red)((x + 2)) to eliminate the fraction while keeping the equation balanced:
color(red)((x + 2))[(x^2 + 5x)/(x + 2)] = color(red)((x + 2))(-10x - 24)
cancel(color(red)((x + 2)))[(x^2 + 5x)/color(red)(cancel(color(black)(x + 2)))] = (color(red)(x) * -10x) + (color(red)(x) * -24) + (color(red)(2) * -10x) + (color(red)(2) * -24)
x^2 + 5x = -10x^2 - 24x - 20x - 48
x^2 + 5x = -10x^2 - 44x - 48
Then, add color(red)(10x^2) and color(blue)(44x) and color(orange)(48) to each side of the equation to put the equation in standard form:
x^2 + color(red)(10x^2) + 5x + color(blue)(44x) + color(orange)(48) = -10x^2 + color(red)(10x^2) - 44x + color(blue)(44x) - 48 + color(orange)(48)
1x^2 + color(red)(10x^2) + 5x + color(blue)(44x) + color(orange)(48) = 0 - 0 - 0
(1 + color(red)(10))x^2 + (5 + color(blue)(44))x + 48 = 0
11x^2 + 49x + 48 = 0
We can now use the quadratic equation to solve this problem:
The quadratic formula states:
For color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0, the values of x which are the solutions to the equation are given by:
x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))
Substituting:
color(red)(11) for color(red)(a)
color(blue)(49) for color(blue)(b)
color(green)(48) for color(green)(c) gives:
x = (-color(blue)(49) +- sqrt(color(blue)(49)^2 - (4 * color(red)(11) * color(green)(48))))/(2 * color(red)(11)
x = (-color(blue)(49) +- sqrt(2401 - 2112))/22
x = (-color(blue)(49) +- sqrt(289))/22
x = (-color(blue)(49) - 17)/22 and x = (-color(blue)(49) + 17)/22
x = -66/22 and x = -32/22
x = -3 and x = -16/11
