Question

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

Explanation:

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

We have the equations \(xy = 42\) and \(x + y=13\). From \(x + y = 13\), we get \(y = 13 - x\).

Step2: Substitute \(y = 13 - x\) into \(xy = 42\).

We get \(x(13 - x)=42\), which expands to \(13x-x^{2}=42\), or \(x^{2}-13x + 42 = 0\).

Step3: Factor the quadratic equation.

For the quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b=-13\), \(c = 42\)), we need to find two numbers that multiply to \(ac=42\) and add up to \(b=-13\). The numbers are \(-6\) and \(-7\). So \(x^{2}-13x + 42=(x - 6)(x - 7)=0\).

Step4: Solve for \(x\).

Setting each factor equal to zero gives \(x-6 = 0\) or \(x - 7=0\). So \(x = 6\) or \(x = 7\). When \(x = 6\), \(y=13 - 6 = 7\); when \(x = 7\), \(y=13 - 7 = 6\).

Answer:

6, 7