Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Roots of quadratic equatiobn
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Roots of quadratic equatiobn
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Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 1
Theme: Pure Mathematics
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Find sum and products of roots of quadratic equation Form quadratic equation given sum and products of roots State condition for quadratic equation to have equal roots, real roots and no roots Identify related conditions for given line to in tersect a given curve, be tangent to curve, not in tersect curve Solve various types of problems on roots of quadratic equations
This section provides a detailed breakdown of the theoretical foundations and practical methods for working with roots of quadratic equations. A. The Standard Form of a Quadratic Equation A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. The values of $x$ that satisfy this equation are called its roots, solutions, or zeros. B. Sum and Product of Roots Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. From the quadratic formula, the roots are given by: $\alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$ $\beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$ Derivation of Sum of Roots: $\alpha + \beta = \left(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\right) + \left(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\right)$ $\alpha + \beta = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a}$ $\alpha + \beta = \frac{-2b}{2a}$ $\boxed{\text{Sum of roots } (\alpha + \beta) = -\frac{b}{a}}$ Derivation of Product of Roots: $\alpha \beta = \left(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\right) \times \left(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\right)$ This is of the form $(P+Q)(P-Q) = P^2 - Q^2$, where $P = -b$ and $Q = \sqrt{b^2 - 4ac}$. $\alpha \beta = \frac{(-b)^2 - (\sqrt{b^2 - 4ac})^2}{(2a)^2}$ $\alpha \beta = \frac{b^2 - (b^2 - 4ac)}{4a^2}$ $\alpha \beta = \frac{b^2 - b^2 + 4ac}{4a^2}$ $\alpha \beta = \frac{4ac}{4a^2}$ $\boxed{\text{Product of roots } (\alpha \beta) = \frac{c}{a}}$ Worked Example 1: Find the sum and product of the roots of the equation $3x^2 - 7x + 2 = 0$.
Solution: Comparing $3x^2 - 7x + 2 = 0$ with $ax^2 + bx + c = 0$, we have: $a = 3$, $b = -7$, $c = 2$. Sum of roots $(\alpha + \beta) = -\frac{b}{a} = -\frac{(-7)}{3} = \frac{7}{3}$. Product of roots $(\alpha \beta) = \frac{c}{a} = \frac{2}{3}$. C. Forming a Quadratic Equation Given Its Roots or Their Sum/Product If $\alpha$ and $\beta$ are the roots of a quadratic equation, then $(x - \alpha)(x - \beta) = 0$.
Expanding this: $x^2 - \beta x - \alpha x + \alpha \beta = 0$ $x^2 - (\alpha + \beta)x + \alpha \beta = 0$ So, a quadratic equation can be formed using the formula: $\boxed{x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0}$ Worked Example 2: Form a quadratic equation whose sum of roots is $-3$ and product of roots is $\frac{1}{2}$.
Solution: Given: Sum of roots $(\alpha + \beta) = -3$ Product of roots $(\alpha \beta) = \frac{1}{2}$ Using the formula $x^2 - (\text{Sum})x + (\text{Product}) = 0$: $x^2 - (-3)x + \frac{1}{2} = 0$ $x^2 + 3x + \frac{1}{2} = 0$ To clear the fraction, multiply through by 2: $2x^2 + 6x + 1 = 0$.
Worked Example 3: Form a quadratic equation whose roots are $5$ and $-2$.
Solution: Let $\alpha = 5$ and $\beta = -2$. Sum of roots $(\alpha + \beta) = 5 + (-2) = 3$. Product of roots $(\alpha \beta) = 5 \times (-2) = -10$. Using the formula $x^2 - (\text{Sum})x + (\text{Product}) = 0$: $x^2 - (3)x + (-10) = 0$ $x^2 - 3x - 10 = 0$. D. Nature of Roots (The Discriminant) The term $b^2 - 4ac$ from the quadratic formula is called the discriminant, denoted by $\Delta$. Its value determines the nature of the roots of a quadratic equation.
1. If $\Delta > 0$ ($b^2 - 4ac > 0$): The equation has two distinct real roots. This means the graph of $y = ax^2 + bx + c$ intersects the x-axis at two different points.
2. If $\Delta = 0$ ($b^2 - 4ac = 0$): The equation has two equal real roots (or one repeated real root). This means the graph of $y = ax^2 + bx + c$ touches the x-axis at exactly one point (the vertex lies on the x-axis). The roots are $\alpha = \beta = -\frac{b}{2a}$.
3. If $\Delta 0$, the equation has two distinct real roots*. b) For $x^2 - 5x + 10 = 0$: $a = 1, b = -5, c = 10$. $\Delta = b^2 - 4ac = (-5)^2 - If $\Delta = 0$ ($b^2 - 4ac = 0$): The equation has two equal real roots (or one repeated real root). This means the graph of $y = ax^2 + bx + c$ touches the x-axis at exactly one point (the vertex lies on the x-axis). The roots are $\alpha = \beta = -\frac{b}{2a}$.
3. If $\Delta 0$, the equation has two distinct real roots. b) For $x^2 - 5x + 10 = 0$: $a = 1, b = -5, c = 10$. $\Delta = b^2 - 4ac = (-5)^2 - 4(1)(10) = 25 - 40 = -15$. Since $\Delta = -15 0 \quad \text{or} \quad (b - m)^2 - 4a(c - k) > 0}$
2. Line is tangent to the curve (intersects at exactly one point): The discriminant of the new quadratic must be equal to zero: $\boxed{B^2 - 4AC = 0 \quad \text{or} \quad (b - m)^2 - 4a(c - k) = 0}$
3. Line does not intersect the curve: The discriminant of the new quadratic must be less than zero: $\boxed{B^2 - 4AC 0$" or "$\Delta > 0$") (2 marks)
Total: 2 marks
5. Regarding the quadratic equation $4x^2 + 4x + 10 = 0$: a) Find the actual sum and product of its roots. b) Given that the sum of roots of another quadratic equation is $\frac{1}{2}$ and its product is $2$, form that quadratic equation. c) Find the value(s) of $x$ (the roots) for the equation $4x^2 + 4x + 10 = 0$. Teacher's
Note: The original evaluation guide question "If sum and product of roots of 4x2 +4x + 10 = 0 are 1⁄2 and
2. Find value of x" contains inconsistent information. The actual sum and product of the given equation are different from 1/2 and
2. This expanded version clarifies the intent for assessment.
Marking Scheme: Part a): Calculates actual sum of roots ($-\frac{4}{4} = -1$) (1 mark) Calculates actual product of roots ($\frac{10}{4} = \frac{5}{2}$) (1 mark)
Part b): Correctly applies formula $x^2 - (\text{Sum})x + (\text{Product}) = 0$ (1 mark) Forms the equation $x^2 - \frac{1}{2}x + 2 = 0$ or $2x^2 - x + 4 = 0$ (1 mark)
Part c): Calculates the discriminant of $4x^2 + 4x + 10 = 0$ (1 mark) Recognises that roots are not real (or proceeds with quadratic formula to find complex roots) (1 mark) Applies quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (1 mark) Correctly finds roots $x = \frac{-4 \pm \sqrt{-144}}{8} = \frac{-4 \pm 12i}{8} = -\frac{1}{2} \pm \frac{3}{2}i$ (1 mark)
Total: 8 marks Overall Total for Assessment: 20 marks*
A. Remediation (Supporting Struggling Learners): Review Prerequisite Skills: Ensure students have a solid understanding of basic algebraic manipulations, including simplifying fractions, expanding brackets, and solving linear equations. Review solving simple quadratic equations by factorisation and the quadratic formula.
Focus on Formulas: Emphasise memorisation and correct application of the sum of roots ($-\frac{b}{a}$) and product of roots ($\frac{c}{a}$) formulas, and the discriminant ($\Delta = b^2 - 4ac$). Provide a formula sheet during initial practice.
Step-by-Step Approach: Break down complex problems into smaller, manageable steps. For example, for "form a quadratic equation", first ask them to find the sum, then the product, then combine them. Simplified
Examples: Provide extra practice with simpler quadratic equations (e.g., $a=1$) before introducing more complex coefficients.
Visual Aids: Use graphs of parabolas to illustrate the meaning of the nature of roots (intersecting x-axis, touching x-axis, not intersecting x-axis).
Peer Tutoring: Pair struggling students with more proficient classmates for guided practice and explanation.
B. Extension and Enrichment (Challenging High-Achieving Learners): Complex Roots: Introduce problems involving complex roots explicitly, asking students to find such roots when $\Delta < 0$.
Higher Order Identities: Explore more advanced identities involving roots, such as $\alpha^3 + \beta^3$ or $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.
Roots of Higher Degree Polynomials: Briefly introduce Vieta's formulas for cubic or quartic equations, showing how the sum and product relationships extend to higher-degree polynomials.
Parameters in Conditions: Pose problems where students need to find ranges of values for multiple parameters based on the nature of roots or line-curve interactions (e.g., find values of $k$ and $m$ for which a line is tangent to a curve and the curve has real roots).
Geometric Interpretation: Challenge students to derive conditions for the line to pass through the vertex of the parabola, or for two parabolas to be tangent to each other.
Real-world Modelling: Assign project-based learning where students research and attempt to model a real-life scenario (e.g., trajectory of a local sport, cost-benefit analysis of a small business venture) using quadratic equations and interpreting the roots. Pure Mathematics
Optimising Agricultural Yields in Nigeria: Farmers in Nigeria often experiment with different quantities of fertilizer or varying planting densities. If the relationship between a specific input (e.g., quantity of fertilizer) and crop yield follows a quadratic function, understanding the roots can help determine the input levels that lead to zero yield (unproductive farming) or, more importantly, the vertex of the parabola which represents the maximum yield. This helps farmers make informed decisions to maximise their produce, contributing to food security. Design of Communication Antennas and Bridges: Many communication antennae (like satellite dishes used for internet or TV in Nigerian homes and businesses) are parabolic in shape. The design of these parabolas involves quadratic equations to ensure efficient signal reception. Similarly, the structural integrity of arch bridges (e.g., bridges over rivers like the Niger or Benue) often relies on parabolic or semi-elliptic designs, where engineers use quadratic equations to model the forces and determine stability, ensuring safety and durability. Financial Modelling and Business Decisions: In Nigerian markets, businesses may model their profit or cost functions using quadratic equations. For instance, the profit ($P$) from selling a certain number of goods ($x$) might be represented as $P = -ax^2 + bx - c$. Finding the roots (break-even points where profit is zero) helps businesses understand their operational thresholds, while the maximum point of the parabola indicates the production level for maximum profit. This is vital for small and medium enterprises (SMEs) across Nigeria, from artisanal producers to tech startups.