Applications of Algebra

Activity 1: The quadratic formula and its usage.
Collaborative Learning, Experiential Learning, Whole class discussion, and Problembased Learning: Collaborative Learning:
- Learners work in convenient groups/pairs to recall key facts about the quadratic formula 𝑎 ) + 𝑏 + 𝑐 = 0.
- Learners work in groups/pairs to establish that the general quadratic solution for the quadratic equation 𝑎 ) + 𝑏 + 𝑐 = 0 is given as: −𝑏 ± √𝑏 ) − 4𝑎 𝑥 = 2𝑎
Example: Find the solution to the equation 2𝑥 ) − 𝑥 − 4 = 0 correct to 2 decimal place.
Solution: Comparing 𝑎 ) + 𝑏 + 𝑐 = 0 to 2𝑥 ) − 𝑥 − 4 = 0 𝑎 = 2 𝑏 = −1 𝑎 𝑐 = −4 −(−1) ± w(−1) ) − 4(2. −4) 𝑥 = 2(2) 𝑥 = 1.69 𝑜 𝑥 = −1.19
Collaborative Learning, Experiential Learning, Whole class discussion, and Problembased Learning: Collaborative Learning:
- Learners work in convenient groups/pairs to recall key facts about the quadratic formula 𝑎 ) + 𝑏 + 𝑐 = 0.
- Learners use the method of completing the squares to write the general quadratic functions 𝑎 ) + 𝑏 + 𝑐 = 0 in the form
- 𝐴(𝑥 ± 𝐵) ) ± 𝐶 = 0 or 𝐴(𝑥 ± 𝐵) ) = 𝐶, where 𝐴, 𝐵 and 𝐶 are constants, and 𝐶 is the maximum or minimum value of the function, and it occurs at 𝑥 ± 𝐵 = 0 or 𝑥 ± 𝐵
Example: Express 3𝑥 ) − 5𝑥 − 2 = 0 in the form 𝐴(𝑥 ± 𝐵) ) = 𝐶 where A, B and 𝐶 are constants and state the minimum values of the function
. ) Solution: 3𝑥 ) − 5𝑥 − 2 = 0 ⇒ 3 i𝑥 − / j − !) = 0 ,2
,2 The minimum value of the function is − !)
Collaborative Learning, Experiential Learning, Whole class discussion, and Problembased Learning: Problem-based learning: Learners use the discriminant of a quadratic function to establish and determine the nature of the root, solution or zeros of the function. Learners recognise that the expression. 𝑏 ) − 4𝑎 of the general quadratic equation 𝑎 ) + 𝑏 + 𝑐 = 0 determines the nature of the roots of the function, and the following properties hold:
- If 𝑏 ) − 4𝑎 ≥ 0, the roots are real.
- If 𝑏 ) − 4𝑎 > 0, the roots are real and different.
- If 𝑏 ) − 4𝑎 < 0, the roots are complex and imaginary.
- If 𝑏 ) − 4𝑎 = 0, the roots are real and equal, or the equation is a perfect square.
Example: Determine the nature of the roots of the function 𝑎 ) − 6𝑎 + 4 = 0
Solution: 𝑏 ) − 4𝑎 = (−6) ) − 4(1)(4) = 36 - 16 = 20; since 𝑏 ) − 4𝑎 > 0, the roots are real and different.
Activity 1: Use the roots (sum and products) to find other quadratic equations.
Collaborative Learning: Learners work in convenient groups/pairs to explore the relationship between the constants 𝑎, 𝑏 𝑎 𝑐 of the general quadratic function 𝑎 ) + 𝑏 + 𝑐 = 0 and 𝛼 𝑎 𝛽, and use these relations to write other quadratic equations with given roots.
Example 1: Form an equation whose roots are 𝑚 and 2.
Solution: Sum of roots = (𝑚 + 2) Products of roots = (2𝑚) Equation: 𝑥 ) − (𝑠 𝑜 𝑟)𝑥 + (𝑝 𝑜 𝑟) = 0 𝑥 ) − (𝑚 + 2)𝑥 + (2𝑚) = 0
Example 2: If 𝛼 𝑎 𝛽 are the roots of the equation 3𝑥 ) − 𝑥 − 3 = 0, find the value of 𝛼 ) − 𝛽 ) , if 𝛼 > 𝛽,
Solution: ! The sum of roots (𝛼 + 𝛽)= " Product of roots (𝛼. 𝛽) = −1
! 𝛼 ) − 𝛽 ) = ( 𝛼 + 𝛽)(𝛼 − 𝛽) = 2 √37
Teaching and Learning Resources:
- Cut out shapes of different colours and orientation
- Maths Technology learning apps, tools and devices
- Chalkboard illustrations
- Worksheets
- Cut-out geometrical shapes of different colours and orientation
Assessment (1.1.2.AS.13). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.