Question

write two numbers that multiply to the value on top and add to the value on bottom. answer attempt 1 out of 2

Explanation:

Step1: Let the two - numbers be \(x\) and \(y\).

We have the equations \(xy=\text{top number}\) and \(x + y=\text{bottom number}\). Let the top number be \(a\) and the bottom number be \(b\). From \(y=\frac{a}{x}\), substitute into \(x + y=b\) gives \(x+\frac{a}{x}=b\), or \(x^{2}-bx + a = 0\). Here \(a = 12\) and \(b = 7\), so the quadratic equation is \(x^{2}-7x + 12=0\).

Step2: Factor the quadratic equation.

We know that \(x^{2}-7x + 12=(x - 3)(x - 4)=0\) (since \((x - m)(x - n)=x^{2}-(m + n)x+mn\) and we need \(m + n=7\) and \(mn = 12\)).

Step3: Solve for \(x\).

Setting \((x - 3)(x - 4)=0\), we get \(x=3\) or \(x = 4\). When \(x = 3\), \(y = 4\) and when \(x = 4\), \(y = 3\).

Answer:

3, 4