Question

What is added to get -4 but multiplied to get 3

Answer 1

- 1 and - 3

Explanation:

- 1 + (- 3) = - 1 - 3 = - 4

and

- 1 × - 3 = 3

Answer 2

-1 and -3

Explanation:

To solve this problem, we're looking for two numbers: one that, when added together, gives us a sum of -4, and when multiplied together, gives a product of 3.

Let's break down the problem:

Numbers that add to -4: To find two numbers that add up to -4, we can set up an equation using variables. Let's call the two numbers "x" and "y." So, we have:

⇒ x + y = -4

Numbers that multiply to 3: Similarly, we can set up an equation for the product of the two numbers:

⇒ x · y = 3

Now, we have a system of two equations with two variables. We can solve for "x" and "y" using different methods, such as substitution or elimination. However, let's solve it using substitution:

From the first equation (x + y = -4), we can express one variable in terms of the other. Let's solve for "x":

⇒ x + y = -4

∴ x = -4 - y

Now substitute this value of "x" into the second equation (x · y = 3):

⇒ x · y = 3

⇒ ( -4 - y) · y = 3

⇒ y( -4 - y) = 3

Simplify the equation:

⇒ -4y - y² = 3

Rearrange the equation to a quadratic form:

⇒ y² + 4y - 3 = 0

Now we have a quadratic equation in terms of "y." We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's factor the quadratic:

⇒ (y + 3)(y - 1) = 0

This gives us two possible values for "y": y = -3 or y = 1.

Now, we can use these values to find the corresponding values of "x" using the equation x = -4 - y:

For y = -3:

x = -4 - (-3) = -1

For y = 1:

x = -4 - 1 = -5

So, the two pairs of numbers are (-1, -3) and (-5, 1). Let's check each pair to assure they satisfy the conditions.

For (-5, 1):

-5 + 1 = -4 ✓

-5 · 1 = -5 ≠ 3

For (-1, -3):

-1 + -3 = -4 ✓

-1 · -3 = 3 ✓

Thus, the correct pair that satisfies the condition is (-1, -3).

`\hrulefill`

Alternative Method (Easier):

There's an easier alternative method that involves thinking about factors of 3 and their signs. Let's walk through this approach:

We're looking for two numbers that multiply to give 3 and add up to give -4. Let's figure out what pairs of numbers multiply to get 3.

Pairs of numbers that multiply to get 3:

• 1 and 3
• -1 and -3

Only one of these pairs adds up to -4. That pair would be -1 and -3.

This method can be quicker for cases where the factors and signs are relatively simple, as in this case. However, for more complex numbers or equations, using algebraic methods can still be quite useful.