Factorise, solve and reconstruct a rational quadratic

Profit rule

\(\displaystyle P(n)=2(n-3)(n-11)\)

Break-even condition

\(\displaystyle P(n)=0\)

A workshop models profit in hundreds of dollars, where n is hundreds of items made. Break-even means the profit is zero. What are the two break-even production levels?

Move the r slider through its values and watch the intercepts, the vertex and the readouts. Then leave r fixed and move the a slider through its values and watch the same features

Move r, then leave r fixed and move a. What happened at every position of the sliders?

Roots, factors and zeros

A factor of the form n minus r gives the root n = r and a zero of the graph. Put the equation in zero form to read the roots from the factors, and build x minus r factors with any leading scale to reconstruct the rule

1. 1Use zero form Move every term to one side so a product equals zero before solving \(\displaystyle P(n)=0\Rightarrow 2(n-3)(n-11)=0\)
2. 2Read factor signs A factor n minus r gives the root r, so the sign reverses when the factor is set to zero \(\displaystyle n-3=0\Rightarrow n=3,\quad n-11=0\Rightarrow n=11\)
3. 3Apply the null factor law If a product equals zero, set each variable factor to zero in turn \(\displaystyle 2(n-3)(n-11)=0\Rightarrow n=3\quad\text{or}\quad n=11\)
4. 4Report each root once List each distinct root a single time. Each root r rebuilds as the factor (x minus r), kept with any leading scale \(\displaystyle n=3,\;n=11\Rightarrow P(n)=2(n-3)(n-11)\)

Original profit rule

\(\displaystyle P(n)=3n(n-8)\)

Original break-even condition

\(\displaystyle P(n)=0\)

New break-even levels

\(\displaystyle n=2\text{ and }n=10\)

Leading coefficient after the change

\(\displaystyle 3\)

A workshop models profit in hundreds of dollars, where n is hundreds of items and cannot be negative. After a pricing change, the break-even levels become 2 and 10 with the same leading coefficient 3. What were the original break-even values?

Set each variable factor to zero in turn

Original profit rule

\(\displaystyle P(n)=3n(n-8)\)

Original break-even condition

\(\displaystyle P(n)=0\)

New break-even levels

\(\displaystyle n=2\text{ and }n=10\)

Leading coefficient after the change

\(\displaystyle 3\)

A workshop models profit in hundreds of dollars, where n is hundreds of items made. After a pricing change, write the new factorised profit rule with leading coefficient 3 and break-even levels n = 2 and n = 10.

Turn each new break-even level into a factor of the form n minus the root, then keep the leading 3 in front

Using the null factor law too early. One of these lines is right. Which one?