Goals
1. Identify and solve quadratic equations using Bhaskara's formula.
2. Use the sum and product methods to solve quadratic equations.
3. Understand how quadratic equations apply to real-world problems.
Contextualization
Quadratic equations pop up in many aspects of our everyday lives and across various professions. They come into play when calculating the path of a thrown object, forecasting financial gains or losses in businesses, or in engineering to assess the strength of materials. Grasping these equations is vital for tackling and resolving intricate problems.
Subject Relevance
To Remember!
Identifying Quadratic Equations
A quadratic equation is a polynomial equation of degree two, widely expressed in the format ax² + bx + c = 0, where a, b, and c are real numbers and a cannot be zero. Recognising a quadratic equation is key to applying the correct methods to solve them.
-
Standard form: ax² + bx + c = 0
-
'a' must be non-zero
-
The equation can yield up to two distinct real solutions
Solving with Bhaskara's Formula
Bhaskara's formula is an effective way to solve quadratic equations. The formula x = (-b ± √(b² - 4ac)) / 2a helps find the roots of the equation by determining the values of x that satisfy it.
-
Formula: x = (-b ± √(b² - 4ac)) / 2a
-
The discriminant (b² - 4ac) indicates the number of real solutions
-
If the discriminant is positive, expect two distinct real solutions; if it's zero, expect one solution; if negative, there are no real solutions
Sum and Product Methods
The sum and product methods serve as alternatives to Bhaskara's formula for solving quadratic equations. They leverage the relationships between the roots of the equation, summarising the sum of the roots as -b/a and the product of the roots as c/a.
-
Sum of the roots: -b/a
-
Product of the roots: c/a
-
Practical for equations that are easy to factor
Practical Applications
-
Calculate the path of a kicked soccer ball using quadratic equations to estimate its landing spot.
-
Forecast profits and losses in a business by modeling profits as a quadratic function to pinpoint maximum profit opportunities.
-
Engineer structures like bridges and buildings using quadratic equations to guarantee strength and safety.
Key Terms
-
Quadratic Equation: A polynomial equation of degree two in the form ax² + bx + c = 0.
-
Bhaskara's Formula: A method for solving quadratic equations, expressed as x = (-b ± √(b² - 4ac)) / 2a.
-
Discriminant: Part of Bhaskara's formula (b² - 4ac) that indicates the number of real solutions of a quadratic equation.
-
Sum of the Roots: The summation of the solutions of a quadratic equation, defined by -b/a.
-
Product of the Roots: The multiplication of the solutions of a quadratic equation, defined by c/a.
Questions for Reflections
-
How might quadratic equations aid you in solving complex problems in your future career?
-
Why is comprehending the discriminant crucial when working through quadratic equations?
-
What additional fields outside of mathematics involve quadratic equations, and what impact does this have on our society?
Mini Challenge: Planning a Trajectory
In this mini-challenge, you'll put your knowledge of quadratic equations into practice by planning the trajectory of a projectile using a simple catapult.
Instructions
-
Form groups of 3 to 4 learners.
-
Utilise the provided resources (rubber bands, plastic spoons, cardboard, etc.) to construct a basic catapult.
-
Calculate the projectile's trajectory with quadratic equations and predict its landing spot.
-
Document all calculations and predictions.
-
Test the catapult and compare actual outcomes with theoretical predictions.
-
Engage in group discussions to explore potential reasons for any discrepancies between practical and theoretical results, and propose improvements.
