Question

Решите задачу с помощью системы уравнений. Разность двух чисел равна 8, а их произведение 240. Найти эти числа ​

Answer 1

Ответ:

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Answer 2

Problem Analysis

To solve this problem using a system of equations, we need to set up two equations based on the given information. Let's assume the two numbers are x and y.

1. The difference between the two numbers is 8: x - y = 8. 2. The product of the two numbers is 240: x * y = 240.

We can solve this system of equations to find the values of x and y.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.

From equation 1, we can express x in terms of y: x = y + 8.

Substituting this value of x into equation 2, we get:

(y + 8) * y = 240.

Expanding the equation, we have:

y^2 + 8y = 240.

Rearranging the equation, we get:

y^2 + 8y - 240 = 0.

Now we have a quadratic equation in terms of y. We can solve this equation to find the possible values of y. Once we have the value(s) of y, we can substitute them back into equation 1 to find the corresponding value(s) of x.

Solving the Quadratic Equation

To solve the quadratic equation y^2 + 8y - 240 = 0, we can factorize it or use the quadratic formula. Let's use the quadratic formula.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a).

In our case, a = 1, b = 8, and c = -240. Substituting these values into the quadratic formula, we get:

y = (-8 ± √(8^2 - 4 * 1 * -240)) / (2 * 1).

Simplifying further, we have:

y = (-8 ± √(64 + 960)) / 2.

y = (-8 ± √1024) / 2.

y = (-8 ± 32) / 2.

This gives us two possible values for y:

1. y = (-8 + 32) / 2 = 24 / 2 = 12. 2. y = (-8 - 32) / 2 = -40 / 2 = -20.

Finding the Corresponding Values of x

Now that we have the values of y, we can substitute them back into equation 1 to find the corresponding values of x.

1. For y = 12: x - 12 = 8. Solving for x, we get: x = 8 + 12 = 20.

2. For y = -20: x - (-20) = 8. Solving for x, we get: x = 8 - (-20) = 28.

Therefore, the two numbers that satisfy the given conditions are 20 and 12, and 28 and -20.

Answer

The two numbers that have a difference of 8 and a product of 240 are 20 and 12, and 28 and -20.