Question 1030563
your equation is x * (x + 5) = 24


you first need to simplify the equation by applying the distributive law of multiplication to get:


x^2 + 5x = 24


you then need to subtract 24 from both sides of the equation to get:


x^2 + 5x - 24 = 0


you then need to factor this equation to get:


(x + 8) * (x - 3) = 0


now that you have the factors equal to 0, you can solve for:


x + 8 = 0 which results in x = -8


x - 3 = 0 which results in x = 3.


why can you do this?


because 0 * (x-3) = 0 and (x + 8) * 0 = 0


your original equation was x * (x-5) = 24


you can't solve it the way you solved the factored quadratic equation because the result is not 0.


x * (x+5) = 24 won't work.


if it was x * (x+5) = 0, then it would work.


but it's not.


you could subtract 24 from both sides of the equation to get:


x * (x+5) - 24 = 0


but you still have to simplify it by using the distributive law of multiplication to get:


x^2 + 5x - 24 = 0 and then factor that to get (x+8) * (x-3) = 0.


only then can you separate out the factors and set them equal to 0 and then solve.


let's assume you could do x * (x+5) = 24 and split it up like:


x = 24
x + 5 = 24



you would get x = 24 or x = 19


you would then need to confirm if these solutions were correct by replacing x in the original equations.


when x = 24, x * (x+5) = 24 becomes 24 * 29 = 24 which is clearly not true.


when x = 19, x * (x+5) = 24 becomes 19 * 24 = 24 which is clearly not true again.

 
solving it the right way, you get x = -8 or x = 3


when x = -8, x * (x+5) = 24 becomes -8 * ( -8 + 5) = 24 which becomes -8 * -3 = 24 which becomes 24 = 24 which is true.


when x = 3, x * (x+5) = 24 becomes 3 * (3 + 5) = 3 * 8 = 24 which becomes 24 = 24 which is also true.


breaking up the factors into two separate equations only works when the result is equal to 0.


x * (x+5) = 0 would work, but x * (x+5) = 24 will not.


you needed to convert the equation into its quadratic equivalent and then you need to set that equation equal to 0 and then you needed to factor it to get your answer.


x * (x+5) = 24 becomes x * (x+5) - 24 = 0 which becomes x^2 + 5x - 24 = 0 which is then factored to be (x+8) * (x-3) = 0.


only then could you set x+8 = 0 and x-3 = 0 to solve for x.