Question

How to solve the following with step by step explanations:

find two numbers whose product is six and their sum is -7

Answer 1

Explanation:

To solve the problem of finding two numbers whose product is six and their sum is -7, we can use a system of equations. Let's denote the two numbers as x and y.

Step 1: Set up the equations.

We can set up two equations based on the given conditions:

Equation 1: x * y = 6 (the product is six)

Equation 2: x + y = -7 (the sum is -7)

Step 2: Solve for one variable in terms of the other.

Let's solve Equation 2 for one variable (let's choose y) in terms of the other variable (x).

From Equation 2, we can rewrite it as:

y = -7 - x

Step 3: Substitute the expression for one variable into the other equation.

Now, substitute the expression for y in terms of x into Equation 1.

x * (-7 - x) = 6

Step 4: Simplify and solve the equation.

Expand the equation by multiplying x with -7 and -x:

-7x - x^2 = 6

Rearrange the equation:

x^2 + 7x + 6 = 0

Now, we have a quadratic equation. We can factor it or use the quadratic formula to solve for x. In this case, the equation can be factored:

(x + 1)(x + 6) = 0

Setting each factor equal to zero gives us two possible solutions:

x + 1 = 0 or x + 6 = 0

Solving for x in each case:

x = -1 or x = -6

Step 5: Find the corresponding values of y.

For each value of x, substitute it back into Equation 2 to solve for y.

When x = -1:

y = -7 - (-1) = -7 + 1 = -6

When x = -6:

y = -7 - (-6) = -7 + 6 = -1

Step 6: Check the solutions.

To check if the solutions are correct, substitute the values of x and y into Equation 1 and Equation 2 to see if they satisfy the conditions.

For x = -1 and y = -6:

Equation 1: (-1) * (-6) = 6 (True)

Equation 2: (-1) + (-6) = -7 (True)

For x = -6 and y = -1:

Equation 1: (-6) * (-1) = 6 (True)

Equation 2: (-6) + (-1) = -7 (True)

Both solutions satisfy the given conditions, so the two numbers that have a product of six and a sum of -7 are -1 and -6.

Answer 2

-1 and -6

Explanation:

To find two numbers who product is 6 and their sum is -7, begin by listing the all the pairs of numbers that multiply to 6.

The factors of 6 are 1, 2, 3 and 6.

Recall that the product of two negative numbers is positive.

Therefore, the pairs of numbers that multiply to 6 are:

`\begin{aligned}-1 \;\textsf{and}\; - 6:&\quad &(-1) * (-6) &=6\\1 \;\textsf{and}\;6:&\quad &1 * 6&=6\\-2 \;\textsf{and}\;-3:&\quad& (-2) * (-3) &=6\\2 \;\textsf{and}\;3:&\quad &2 * 3 &=6\end{aligned}`

The sum of each pair of numbers is:

`\begin{aligned}-1 \;\textsf{and}\; - 6:&\quad &(-1) +(-6) &=-7\qquad \leftarrow \sf Sum\;is\;-7\\1 \;\textsf{and}\;6:&\quad &1 +6&=7\\-2 \;\textsf{and}\;-3:&\quad& (-2) +(-3) &=-5\\2 \;\textsf{and}\;3:&\quad &2 +3 &=5\end{aligned}`

Therefore, the two numbers who product is 6 and their sum is -7 are:

• -1 and -6