Goals
1. Get to grips with the coefficients of quadratic equations.
2. Work out the sum and product of the roots using the coefficients.
3. Calculate values that can be expressed as functions of the sum and product of the roots.
Contextualization
Quadratic equations pop up in a variety of everyday scenarios, from the path of a ball being thrown to how we calculate areas and volumes in civil engineering. For instance, when constructing a bridge, engineers rely on quadratic equations to figure out the curve of arches, making sure the structure is stable and safe. Grasping their coefficients and their impact on the solutions of the equation is key to tackling practical challenges and honing solid mathematical reasoning.
Subject Relevance
To Remember!
Coefficients in Quadratic Equations
Coefficients in a quadratic equation are the numerical values that go with the variables. In the standard form ax² + bx + c = 0, 'a', 'b', and 'c' are the coefficients. The coefficient 'a' corresponds to the x² term, 'b' is linked to the x term, and 'c' is the constant term.
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The coefficients shape the look of the parabola shown by the equation.
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The coefficient 'a' affects how the parabola opens; if 'a' is positive, it opens upwards, while if negative, it opens downwards.
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The coefficient 'b' shifts the parabola horizontally, and 'c' indicates where the parabola crosses the y-axis.
Sum of the Roots
You can easily calculate the sum of the roots of a quadratic equation directly from its coefficients. For the equation ax² + bx + c = 0, if the roots are r₁ and r₂, then the sum of the roots (r₁ + r₂) equals -b/a.
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The sum of the roots is a quick way to check if the solutions of an equation match up.
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This property simplifies solving equations and allows for finding solutions without having to solve the whole equation.
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Understanding this relationship is crucial for analysing equations in real-world contexts like engineering and economics.
Product of the Roots
The product of the roots of a quadratic equation can also be determined from the coefficients. For ax² + bx + c = 0, the product of the roots (r₁ * r₂) is equal to c/a.
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Like the sum of the roots, the product provides a quick way to verify solutions.
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It's especially useful for optimisation and modelling tasks where the multiplication of solutions matters.
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Understanding this relationship aids in grasping how quadratic equations behave and their wide-ranging applications.
Practical Applications
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Civil Engineering: Using quadratic equations to calculate the curvature of bridges and arches, ensuring safety and stability.
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Economics: Modeling financial market behaviours, maximising profits, and minimising costs.
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Physics: Studying parabolic motion, like projectile paths, and determining points of maximum range and height.
Key Terms
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Coefficient: A numerical value that multiplies the variable in an equation. In ax² + bx + c = 0, 'a', 'b', and 'c' are the coefficients.
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Roots: The solutions of the quadratic equation, also referred to as zeros or the points where the parabola meets the x-axis.
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Sum of the Roots: The value obtained by adding the solutions of the quadratic equation, calculated as -b/a.
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Product of the Roots: The value obtained by multiplying the solutions of the quadratic equation, calculated as c/a.
Questions for Reflections
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How can understanding the coefficients in a quadratic equation enhance your ability to solve real-world problems moving forward?
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In what ways could the sum and product of the roots streamline problem-solving in fields like engineering and economics?
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Think of a project or scenario in your life that could be modelled using a quadratic equation. How would you apply what you've learned?
Exploring the Path of a Projectile
This mini-challenge is designed to apply your understanding of quadratic coefficients in a real-life context, using concepts from physics and maths to analyse a projectile's path.
Instructions
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Form pairs or small groups.
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Use a simulation tool (like PhET Interactive Simulations) or a physics app to launch a projectile and gather trajectory data.
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Take note of the maximum height and total distance traversed by the projectile.
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Create a quadratic equation that represents the observed path, pinpointing the coefficients a, b, and c.
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Compute the sum and product of the roots of this equation.
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Discuss how the coefficients influence the projectile’s path and compare your findings with the simulated data.
