what two numbers multiply to -42 and add to 19
The Correct Answer and Explanation is:
ChatGPT said:
To find two numbers that multiply to -42 and add to 19, we begin by identifying the properties of the numbers:
- Their product is -42, which means one number is positive and the other is negative (because a negative times a positive gives a negative).
- Their sum is positive 19, which means the positive number is greater in magnitude than the negative one.
Let’s find all factor pairs of -42:
- (-1, 42) → -1 + 42 = 41
- (-2, 21) → -2 + 21 = 19 ✔
- (-3, 14) → -3 + 14 = 11
- (-6, 7) → -6 + 7 = 1
- (1, -42) → 1 + (-42) = -41
- (2, -21) → 2 + (-21) = -19
- (3, -14) → 3 + (-14) = -11
- (6, -7) → 6 + (-7) = -1
Among these, the pair (-2, 21) satisfies both conditions:
-2 × 21 = -42 and -2 + 21 = 19
Textbook-Style Explanation
When solving a problem that asks for two numbers based on their product and sum, it helps to follow a structured approach. Begin by identifying the target product and the target sum. In this case, we are looking for two numbers whose product is -42 and sum is 19.
Since the product is negative, it indicates that one number is positive and the other is negative. A positive number multiplied by a negative number always yields a negative result. The fact that their sum is positive suggests that the positive number must be larger in magnitude than the negative one.
Next, list all the factor pairs of 42. These pairs are combinations of numbers that multiply to 42. Then, assign a negative sign to one number in each pair to explore possible combinations whose product is -42. Check each pair to see if their sum equals 19.
When we test the pair (-2, 21):
- The product is -2 × 21 = -42
- The sum is -2 + 21 = 19
This pair satisfies both conditions. Therefore, the two numbers we are looking for are -2 and 21.
This method of factoring is commonly used in algebra, especially when solving quadratic equations of the form ax² + bx + c = 0, where identifying such pairs helps factor the trinomial into simpler expressions.
