Welcome to Equations and Inequalities!
Hi there! In this chapter, we are going to master the art of finding the "unknown." Whether it's finding the exact point where two paths cross or figuring out a range of values where a business stays profitable, equations and inequalities are your essential tools. Don't worry if algebra has felt like "alphabet soup" in the past—we’re going to break it down step-by-step into clear, logical pieces.
Prerequisite Check: Before we dive in, remember that a quadratic equation usually looks like \(ax^2 + bx + c = 0\). The values of \(a\), \(b\), and \(c\) are just numbers that tell us the shape and position of the curve!
1. The "Nature" of Roots: Using the Discriminant
Sometimes, we don't need to solve the whole equation; we just want to know how many solutions (roots) exist. For this, we use a "detective tool" called the discriminant, denoted by the symbol \(D\) or \(\Delta\).
The formula for the discriminant is: \(D = b^2 - 4ac\)
How to interpret the Discriminant:
- Two Real and Distinct Roots: If \(b^2 - 4ac > 0\), the graph cuts the x-axis at two different points.
- Two Equal Roots (or One Repeated Root): If \(b^2 - 4ac = 0\), the graph just touches the x-axis at one point. It's like a ball bouncing off the floor!
- No Real Roots: If \(b^2 - 4ac < 0\), the graph never touches the x-axis. It is either floating above it or buried below it.
Quick Review Box:
2 Roots \(\rightarrow D > 0\)
1 Root \(\rightarrow D = 0\)
No Roots \(\rightarrow D < 0\)
Common Mistake: Students often forget that "Real Roots" (without the word 'distinct') means the discriminant can be either greater than zero OR equal to zero. So, for "real roots," use \(b^2 - 4ac \geq 0\).
2. Always Positive or Always Negative Quadratics
Have you ever seen a graph that stays strictly on one side of the x-axis? In H1 Math, we call these definite quadratics.
Condition for "Always Positive":
For \(ax^2 + bx + c\) to be always positive (above the x-axis for all \(x\)):
1. The graph must be "Happy" (U-shaped): \(a > 0\)
2. The graph must never touch the x-axis: \(b^2 - 4ac < 0\)
Condition for "Always Negative":
For \(ax^2 + bx + c\) to be always negative (below the x-axis for all \(x\)):
1. The graph must be "Sad" (n-shaped): \(a < 0\)
2. The graph must never touch the x-axis: \(b^2 - 4ac < 0\)
Memory Aid: Notice that in both cases, \(b^2 - 4ac\) must be less than zero because the graph isn't allowed to touch the x-axis!
Key Takeaway: If a quadratic is "always [something]," it means it has no roots, so \(D < 0\).
3. Solving Simultaneous Equations
In this section, we look for the intersection where a straight line (linear) meets a curve (quadratic). The most reliable method here is Substitution.
Step-by-Step Process:
- Isolate: Pick the linear equation and make one variable (either \(x\) or \(y\)) the subject. (e.g., \(y = 2x + 3\)).
- Substitute: Plug this expression into the quadratic equation.
- Simplify: Expand the brackets and move everything to one side to get a standard quadratic equation (\(ax^2 + bx + c = 0\)).
- Solve: Solve for the first variable.
- Find the Partner: Plug your answers back into the linear equation to find the corresponding values of the other variable.
Example: If you have \(y = x + 1\) and \(x^2 + y^2 = 5\), substitute the first into the second: \(x^2 + (x+1)^2 = 5\). Now it’s just a normal quadratic to solve!
4. Solving Inequalities
Inequalities are like equations, but instead of finding a single point, we are finding a range or a "zone."
Analytical Method (The Factorizing Way):
To solve \(ax^2 + bx + c > 0\):
1. Factorize the quadratic to find the critical values (the roots).
2. Sketch a quick mini-graph.
3. If the inequality is \(> 0\), look for the parts of the curve above the x-axis ("the wings").
4. If the inequality is \(< 0\), look for the part of the curve below the x-axis ("the valley").
Graphical Method:
If you are given a graph of \(y = f(x)\) and asked to solve \(f(x) \leq k\):
1. Draw a horizontal line at \(y = k\).
2. Identify the regions where the curve is underneath or touching that line.
3. Write down the \(x\)-values for those regions.
Did you know? You can use your Graphing Calculator (GC) to solve these! Just graph the function and use the 'Zero' or 'Intersect' functions to find your boundaries.
5. Modelling: From Words to Math
In the real world, math doesn't come in neat equations; it comes in stories. Your job is to formulate the equation.
Analogy: Think of this as translating a language. "Is" translates to \(=\), "more than" translates to \(+\), and "product" translates to multiplication.
Tips for Formulating Equations:
- Assign Variables: Clearly state what \(x\) and \(y\) represent (e.g., "Let \(x\) be the number of units sold").
- Look for Totals: Often, one equation will be about the quantity (total items) and the other will be about the value (total cost/profit).
- Check Units: Make sure everything is in the same units (e.g., don't mix cents and dollars!).
Don't worry if this seems tricky at first! Modelling is a skill that improves with practice. Start by underlining key numbers in the word problem.
6. Using the Graphing Calculator (GC)
For H1 Math, your GC is your best friend. For complex equations where you can't factorize easily, you should use the GC to find approximate solutions.
How to find solutions on your GC:
- Go to the Graph mode and enter your equation into \(Y1\).
- Press 2nd + CALC (or G-Solv on Casio).
- Select Zero to find where the graph hits the x-axis.
- Select Intersect if you are finding where two different lines/curves meet.
Key Takeaway: Always ensure your calculator is in the correct "window" so you can actually see where the intersections happen!
Summary Checklist:
- Can I use \(b^2 - 4ac\) to find the number of roots?
- Do I know the conditions for a quadratic to be "always positive"?
- Can I solve linear-quadratic simultaneous equations by substitution?
- Can I solve inequalities by sketching the graph?
- Do I know how to use my GC to find intersection points?
You've got this! Keep practicing the algebraic steps, and soon these patterns will become second nature.