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Grade 10 · Core Mathematics · Lesson 10 of 14

Exact Roots with Signed Factors

A Grade 10 Core Mathematics CBE lesson from the sub-strand Quadratic Expressions and Equations. In it, learners factor a signed quadratic.

Strand Numbers and AlgebraSub-strand Quadratic Expressions and EquationsLesson code 10CM-13-L1040 minutes · 3 slides

KICD learning outcome (f)

Solve quadratic equations by factorisation.

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

  1. Factor a signed quadratic.
  2. Solve both exact linear equations.
  3. Check the middle term and both roots.

Key vocabulary

Monic quadratic
A quadratic whose leading coefficient is one.
s²−390s+29 is monic because its leading coefficient is 1.
Exact root
A root written without rounding or approximation.
The roots 23/19 and −29/7 are exact.
Audit
A systematic check of each important part of a calculation.
An expansion audit confirms 551s−161s=390s for the middle term.

Lesson preview

Slide 1 · Factor the signed expressionOpening · 10 min
A grid method illustrates how to multiply two signed factors to find their expanded quadratic form, detailing the calculation of each term.
Figure 1: A grid method illustrates how to multiply two signed factors to find their expanded quadratic form, detailing the calculation of each term.

A youth group models their poultry project's financial break-even points with a quadratic expression. The factors (19s−23) and (7s+29) combine to a positive leading term, a small positive middle term, and a negative constant. The signed coefficients determine the middle and constant terms. Use Figure 1 to see how these factors combine.

  • Multiply 19s by both terms.
  • Multiply −23 by both terms.
  • Combine 551s and −161s.
  • Check the constant −667.
Worked example. Try (19s−23)(7s+29). The cross terms are 551s and −161s. The constant is −667. Answer: 133s²+390s−667=(19s−23)(7s+29)..

Check question: When expanding the factors (19s−23)(7s+29) to find the youth group's quadratic expression, what is the combined middle term?

  1. 390s
  2. 712s
  3. −390s
  4. −712s
Show answer

390s. Multiply the terms to find the cross terms: 19s×29=551s and −23×7s=−161s. Combining these gives 551s−161s=390s.

Slide 2 · Solve both factorsPractice · 18 min
The zero-product property is applied to set each factor to zero, demonstrating the step-by-step process of solving for the two exact roots.
Figure 2: The zero-product property is applied to set each factor to zero, demonstrating the step-by-step process of solving for the two exact roots.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • 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