The task in the image is asking to find two numbers that meet the following conditions:

1. One number is 57 less than the other.
2. The product of these two numbers is -740.

Let the two numbers be $x$ and $y$, where $x$ is the larger number. According to the problem, we have:

$x = y + 57$

and

$x \cdot y = -740$

Substituting $x = y + 57$ into the second equation:

$(y + 57) \cdot y = -740$

Expanding this:

$y^2 + 57y = -740$

Rearranging the equation:

$y^2 + 57y + 740 = 0$

Now, we solve this quadratic equation. The general formula for solving a quadratic equation $ay^2 + by + c = 0$ is:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $y^2 + 57y + 740 = 0$, $a = 1$, $b = 57$, and $c = 740$. Let's calculate the discriminant and find the values of $y$. math

Coefficients of the quadratic equation

a = 1 b = 57 c = 740

Calculate the discriminant

discriminant = b**2 - 4ac

Calculate the solutions using the quadratic formula

if discriminant >= 0: root1 = (-b + math.sqrt(discriminant)) / (2a) root2 = (-b - math.sqrt(discriminant)) / (2a) else: root1 = root2 = None

(root1, root2)The solutions to the quadratic equation are:

$y = -20 \quad \text{or} \quad y = -37$

Now, we can find the corresponding values for $x$ using $x = y + 57$.

For $y = -20$: $x = -20 + 57 = 37$

For $y = -37$: $x = -37 + 57 = 20$

Thus, the two pairs of numbers are:

1. $37$ and $-20$
2. $20$ and $-37$

These are the two solutions that satisfy the conditions. The first pair is $37$ and $-20$, and the second pair is $20$ and $-37$.

Would you like further clarification on this? Let me know if you have any questions or would like to explore related problems!

Here are 5 related questions:

1. How do you solve a quadratic equation using the quadratic formula?
2. What happens if the discriminant in a quadratic equation is negative?
3. Can the product of two numbers ever be negative if both numbers are positive or both negative?
4. How do you interpret the roots of a quadratic equation in terms of the problem context?
5. How can you verify that the values of $x$ and $y$ satisfy the given conditions in a word problem?

Tip: When solving quadratic equations, always check whether the discriminant is positive, zero, or negative, as this determines the nature of the roots (real or complex).