#1**0 **

Let's do some algebraic manipulation and see what the result yields.

\(2x^2 + 24x - 60 = x(x + 13) \\ 2x^2 +24x - 60 = x^2 + 13x \\ x^2 + 11x - 60 = 0\)

This quadratic happens to be factorable, so I will take advantage of that fact.

\((x - 4)(x+ 15) = 0 \\ x = 4 \text{ or } x = -15\)

Now, just pick the smallest x-value, which is \(x = -15\) in this case.

The3Mathketeers Aug 16, 2023

#2**0 **

To find the smallest possible value of \(x\) that satisfies the equation \(2x^2 + 24x - 60 = x(x + 13)\), we can start by simplifying the right side:

\[x(x + 13) = x^2 + 13x.\]

So, the equation becomes:

\[2x^2 + 24x - 60 = x^2 + 13x.\]

Combine like terms:

\[2x^2 + 24x - 60 - x^2 - 13x = 0.\]

Simplify further:

\[x^2 + 4x - 60 = 0.\]

Now, we have a quadratic equation. To find the solutions for \(x\), we can factor the quadratic or use the quadratic formula:

\[x^2 + 4x - 60 = (x - 6)(x + 10) = 0.\]

Setting each factor equal to zero:

\[x -6 = 0 \quad \text{or} \quad x + 10 = 0.\]

This gives us two possible values for \(x\):

1. \(x = 6\)

2. \(x = -10\)

Since you're looking for the smallest possible value of \(x\), the answer is \(x = \boxed{-10}\).

SpectraSynth Aug 16, 2023

#4**0 **

SpectraSynth, this is a good-faith effort towards solving this problem, but why did the line \(2x^2 + 24x - 60 - x^2 - 13x = 0\) turn into \(x^2 + 4x - 60 = 0\)? Notice how \(24x - 13x \neq 4x\).

The3Mathketeers
Aug 16, 2023

#3**-2 **

We can factor the left side as 2(x2+12x+30). Since the constant term of the quadratic is 30, its factors must add up to 12. The only pair of factors of 30 with this property is 10 and 3. So we can factor the quadratic as 2(x+10)(x+3). Setting this equal to x(x+13), we get x+10=0 or x+3=0. The smaller of these roots is x=−10. Therefore, the smallest possible value of x is −10.

maximum Aug 16, 2023

#5**0 **

maximum, this is a good-faith effort towards solving this problem, but you forgot to consider the right-hand side of the equation. Also, your attempt at factorization is not correct.

If you check your work, you will discover that your claims are not accurate.\(2(x + 10)(x + 3) = 2(x^2 + 10x + 3x + 30) = 2(x^2 + 13x + 30) \neq 2(x^2 + 12x + 30)\).

The3Mathketeers
Aug 16, 2023