what multiplies to -14

2 min read 23-12-2024

Finding numbers that multiply to -14 might seem simple at first, but understanding the process is crucial for solving more complex algebraic equations. This article will explore different methods to find these factor pairs, explaining the concepts clearly and providing examples. Let's dive in!

Understanding Negative Products

The key to solving "what multiplies to -14" lies in understanding how negative numbers work in multiplication. Remember this crucial rule:

A negative number multiplied by a positive number always results in a negative number.

This means that to get a product of -14, we need one positive factor and one negative factor.

Finding the Factor Pairs

To find the pairs of numbers that multiply to -14, we first consider the factors of 14 (ignoring the negative sign for now). The factor pairs of 14 are:

1 and 14
2 and 7

Now, let's introduce the negative sign. Since we need a negative product, one number in each pair must be negative. This gives us the following pairs that multiply to -14:

1 and -14 (1 * -14 = -14)
-1 and 14 (-1 * 14 = -14)
2 and -7 (2 * -7 = -14)
-2 and 7 (-2 * 7 = -14)

Therefore, the pairs of numbers that multiply to -14 are (1, -14), (-1, 14), (2, -7), and (-2, 7).

Visualizing with a Number Line

A number line can be a helpful visual tool. Imagine plotting -14 on a number line. The pairs we found represent distances on either side of zero that, when multiplied, give us -14.

Applying this to Algebra

Understanding how to find factors is essential for solving quadratic equations. For instance, consider the equation:

x² + 3x - 14 = 0

To solve this, we need to find two numbers that add up to 3 (the coefficient of x) and multiply to -14. Looking at our pairs above, 7 and -2 fit the bill (7 + (-2) = 5, not 3, so let's try a different approach). Let's factor the quadratic expression:

(x + 7)(x - 2) = 0

This gives us two possible solutions for x: x = -7 or x = 2.

More Complex Scenarios

The same principles apply to finding factors for larger negative numbers. For example, what multiplies to -36? You would first find the factor pairs of 36, then make one number in each pair negative to achieve a negative product.

Conclusion: Mastering Factor Pairs

Knowing how to find the factors of negative numbers is a fundamental skill in algebra. This article demonstrated how to find all the pairs of numbers that multiply to -14, and explained how to use this knowledge to solve more advanced problems such as factoring quadratic equations. Remember the key: one positive factor and one negative factor to achieve a negative product. Now you're equipped to tackle more challenging factor problems!