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Factorise, solve and reconstruct a rational quadratic

Start a set of 8
Step 4 of 6
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
\(\displaystyle 3\)

The given original profit rule measures profit in hundreds of dollars, with n a non-negative number of hundreds of items. What were the original break-even values?

\[P(n)=0\]
Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Solving by factorisation

A factorised quadratic is solved by writing zero form, applying the null factor law and solving the resulting linear equations. Each root must be checked against the factors, and a repeated root is reported once

\[AB=0\Rightarrow A=0\quad\text{or}\quad B=0\]
  1. Use zero form Move all terms to one side before applying the null factor law \(\displaystyle 2x\left(x-3\right)=4\Longleftrightarrow 2x\left(x-3\right)-4=0\)
  2. Read factor signs Set each variable factor equal to zero; moving its constant to the other side reverses the sign \(\displaystyle x-5=0\Rightarrow x=5,\quad x+2=0\Rightarrow x=-2\)
  3. Apply the null factor law If a product is zero, set each factor equal to zero \(\displaystyle AB=0\Rightarrow A=0\quad\text{or}\quad B=0\)
  4. Report each root once A repeated factor gives one distinct root, while two different factors give two roots \(\displaystyle 2x\left(x-5\right)=0\Rightarrow x=0\quad\text{or}\quad x=5\)

Worked example

Solve the quadratic equation by factorisation

\[3x\left(x-4\right)=0\]
  1. \(\displaystyle 3x\left(x-4\right)=0\)The equation is already in zero form
  2. \(\displaystyle x=0\quad\text{or}\quad x-4=0\)A product is zero when at least one factor is zero
  3. \(\displaystyle x=0\quad\text{or}\quad x=4\)Solve the two linear equations and list both roots

Common mistakes

Using the null factor law too early
✕\(\displaystyle 2x\left(x-3\right)=4\Rightarrow x=0\quad\text{or}\quad x=3\)✓\(\displaystyle 2x\left(x-3\right)=4\Longleftrightarrow 2x\left(x-3\right)-4=0\)
The null factor law applies only to a product equal to zero, so all terms must first be moved to one side
Reversing a factor sign
✕\(\displaystyle x+6=0\Rightarrow x=6\)✓\(\displaystyle x+6=0\Rightarrow x=-6\)
Subtract the positive constant from both sides to isolate the root; its sign is therefore negative
Counting a repeated root twice
✕\(\displaystyle \left(x-2\right)^{2}=0\Rightarrow x=2\quad\text{and}\quad x=2\)✓\(\displaystyle \left(x-2\right)^{2}=0\Rightarrow x=2\text{ with multiplicity }2\)
The repeated factor gives one distinct root, even though that root has multiplicity 2

Difficulty levels

introInteger roots and leading coefficient 1 or −1, including rearrangement to zero form
standardFractional roots or a leading scale whose magnitude differs from 1
challengeA non-zero right side requiring rearrangement or reconstruction from roots
examA factorisation choice followed by ordered root work, including repeated-root cases

Questions about this topic

Why must one side be zero?+

The null factor law states that a product is zero when at least one factor is zero, so it cannot be applied while the product has a non-zero value

How is a root read from a factor?+

Set the factor equal to zero and isolate the variable; a factor with a subtracted constant gives that constant as its root, while an added constant gives its negative

How is a repeated root reported?+

Report the root value once and state its multiplicity if that information is required

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model