Question

write two numbers that multiply to the value on top and add to the value on bottom. answer attempt 2 out of 2 12 + × 35

Explanation:

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

We have the equations \(xy = 35\) and \(x + y=12\). From \(x + y = 12\), we get \(y = 12 - x\). Substitute \(y = 12 - x\) into \(xy = 35\), we get \(x(12 - x)=35\).

Step2: Expand the equation.

\(12x-x^{2}=35\), which can be rewritten as \(x^{2}-12x + 35 = 0\).

Step3: Factor the quadratic equation.

For the quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 1\), \(b=-12\), \(c = 35\)), we need to find two numbers that multiply to \(ac=35\) and add up to \(b=-12\). The numbers are - 5 and - 7. So \(x^{2}-12x + 35=(x - 5)(x - 7)=0\).

Step4: Solve for \(x\).

If \((x - 5)(x - 7)=0\), then \(x=5\) or \(x = 7\). When \(x = 5\), \(y=12 - 5 = 7\); when \(x = 7\), \(y=12 - 7 = 5\).

Answer:

5 and 7