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Formulate and solve a quadratic problem

Start a set of 8
Step 6 of 6
Model equation
\(\displaystyle r^{2}+7r-144=0\)
r is a number of rows
\(\displaystyle r>0\)

Omitting the model domain. One of these lines is right. Which one?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

Formulating and solving a quadratic model

A quadratic model begins with a variable that has a stated meaning and domain. Translate the relation into an equation, solve all mathematical roots and retain only the roots allowed by the situation

\[x\left(x+4\right)=45\Longleftrightarrow x^{2}+4x-45=0\]
  1. Define the variable Give the variable a meaning, unit and domain before forming an equation \(\displaystyle x>0\)
  2. Express dependent quantities Write every quantity in terms of the same variable and keep the stated relation \(\displaystyle x\left(x+4\right)=45\)
  3. Write zero form Move all terms to one side so the quadratic can be solved by factorisation or another exact method \(\displaystyle x\left(x+4\right)=45\Longleftrightarrow x^{2}+4x-45=0\)
  4. Check the context Test every mathematical root against the original sign, unit, whole-number and domain conditions \(\displaystyle x^{2}+4x-45=0\Rightarrow x=5\text{ or }x=-9,\quad x>0\Rightarrow x=5\)

Worked example

A rectangular display area has side lengths x metres and x + 3 metres. Its area is 40 square metres. Both side lengths are positive. Find x.

\[x\left(x+3\right)=40\]
  1. \(\displaystyle x\left(x+3\right)=40\)Use x for one side and x + 3 for the other, so their product gives the area
  2. \(\displaystyle x^{2}+3x-40=0\)Expand the product and move 40 to the left to obtain zero form
  3. \(\displaystyle \left(x+8\right)\left(x-5\right)=0\Rightarrow x=-8\text{ or }x=5\)Factor the quadratic and apply the null factor law to obtain both mathematical roots
  4. \(\displaystyle x>0\Rightarrow x=5\text{ m}\)The negative root cannot be a side length, so the positive root is the meaningful value

Common mistakes

Omitting the model domain
✕\(\displaystyle x^{2}+3x-40=0\quad\Rightarrow\quad x=-8\)✓\(\displaystyle x^{2}+3x-40=0,\quad x>0\quad\Rightarrow\quad x=5\)
The equation may have more than one mathematical root, but the model's domain must still be retained when a meaningful value is chosen
Adding dependent quantities
✕\(\displaystyle x+(x+3)=40\)✓\(\displaystyle x\left(x+3\right)=40\)
The area of a rectangle is the product of its side lengths, so the two dimensions must be multiplied
Changing the stated relation
✕\(\displaystyle x\left(x-3\right)=40\)✓\(\displaystyle x\left(x+3\right)=40\)
The second side is three units greater than x, so its expression is x + 3.

Difficulty levels

introUse a count-product or direct-square model and apply its stated domain
standardInterpret count, dimension and direct-square models using the stated domain; one or both mathematical roots may be feasible
challengeUse count, dimension, direct-square and balance models, including rational coefficients and contexts where both roots are feasible
examModel selection, exact root work and a justification of the feasible result

Questions about this topic

Why must a model include a domain?+

The algebraic equation describes mathematical possibilities, while the domain records which values have the required meaning in the situation

Why are all roots checked in the original context?+

A quadratic can produce a negative, non-integer or otherwise out-of-range root even when the equation is solved correctly

How is a quadratic model formed?+

Define one variable, express the related quantities in terms of it, use the stated product or squared relation, and rearrange to zero form

Can I use a calculator?+

A calculator is allowed unless a question is marked no 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