Question

write two numbers that multiply to the value on top and add to the value on bottom. -15 + × -16

Explanation:

Step1: Let the two - numbers be \(a\) and \(b\).

We know that \(a\times b\) is the product and \(a + b\) is the sum. We need \(ab=- 16\) and \(a + b=-15\). We can consider the quadratic - equation form \(x^{2}+bx + c = 0\), where the roots of the quadratic equation \(x^{2}+15x - 16=0\) are the two numbers we are looking for. The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), but we can also factorize. We need to find two numbers that multiply to \(-16\) (the constant term) and add up to \(15\) (the coefficient of \(x\)). The factors of \(16\) are \(1\times16\), \(2\times8\), \(4\times4\). Since the product is negative and the sum is negative, we consider the pair of numbers such that one is positive and one is negative.

Step2: Factorize the quadratic - like expression.

We want to find two numbers \(m\) and \(n\) such that \(mn=-16\) and \(m + n=-15\). The numbers are \(-16\) and \(1\) because \((-16)\times1=-16\) and \(-16 + 1=-15\).

Answer:

1, - 16