Solving by Factorisation

By the end of this lesson, learners should be able to

1. Factor a non-monic quadratic.
2. Set each factor equal to zero.
3. Verify integer and fractional roots in the original equation.

Key vocabulary

Zero-product property

If a product is zero, at least one factor must be zero.

After factorising, set 17r+13 or r−19 equal to zero.

Root

A value that makes a polynomial equal to zero.

The value r=19 is a root because the factor r−19 vanishes.

Exact value

A value kept in its precise fractional form.

−13/17 is an exact value for the second root.

Lesson preview

Slide 1 · Find the factorsOpening · 10 min

For a 247 m² community forest with width r and length 17r−310, solving 17r²−310r−247=0 ensures responsible land use. This needs factors with product −4199 and sum −310. The target product and middle coefficient determine the pair. Figure 1 shows the area model.

• Multiply 17 by −247 to get −4199.
• Find 13 and −323 with sum −310.
• Split −310r into 13r−323r.
• Group terms for factorisation.

Worked example. Use 17×(−247)=−4199. The pair 13 and −323 sums to −310. 17r²+13r−323r−247=(17r+13)(r−19). Answer: 17r²−310r−247=(17r+13)(r−19)..

Check question: When factorising the quadratic expression $17r^2-310r-247$, why is the target product $-4199$?

1. It is the product of the first coefficient $17$ and the constant term $-247$.
2. It is the sum of the middle coefficient $-310$ and the constant term $-247$.
3. It is the product of the first coefficient $17$ and the middle coefficient $-310$.
4. It is the square of the middle coefficient $-310$.

Show answer

It is the product of the first coefficient $17$ and the constant term $-247$. The target product used to split the middle term in a quadratic expression $a{x}^2+bx+c$ is found by multiplying the leading coefficient $a$ by the constant term $c$. Here, $17\times (-247)=-4199$.

Slide 2 · Solve and check the integer rootPractice · 18 min

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• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 13 lessons in this sub-strand

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All 14 lessons in Quadratic Expressions and Equations

1. 1Building Quadratic Expressions from DimensionsForm quadratic expressions from different situations
2. 2Four Products on a NoticeboardForm quadratic expressions from different situations
3. 3Deriving a Square Identity by AreaDerive the quadratic identities from the concept of area
4. 4Restoring a Difference-Square AreaDerive the quadratic identities from the concept of area
5. 5Nearby Bases for Mental SquaringApply quadratic identities in numerical cases
6. 6Factor Pairs and GroupingFactorise quadratic expressions from different situations
7. 7Forming an Area EquationForm quadratic equations in different situations
8. 8A Consecutive-Integer EquationForm quadratic equations in different situations
9. 9Solving by FactorisationSolve quadratic equations by factorisation
10. 10Exact Roots with Signed FactorsSolve quadratic equations by factorisation
11. 11A Triangular Sign ModelApply quadratic equations to real life situations
12. 12Planning a Rectangular TankApply quadratic equations to real life situations
13. 13Exploring Quadratic Models for a Community MuralExplore the use of quadratic equations in real life situations
14. 14Exploring a Community Greenhouse FrameExplore the use of quadratic equations in real life situations