Quadratic Equations: A Beginner's Guide

 

Imagine you're throwing a ball up in the air. It goes up, reaches a peak, and then falls back down. The path it takes can be described by a quadratic equation. Let's explore what that means:

1. What is a Quadratic Equation?

It's an equation that looks like this: ax² + bx + c = 0. Don't be intimidated by the symbols! Let's break it down:

• x: This represents an unknown value, like the time it takes for the ball to reach the ground.
• a, b, c: These are numbers, like the initial speed of the ball, its starting height, and the effect of gravity.
• ²: This means "squared," or multiplied by itself. For example, x² is the same as x times x.

2. Why are they called "Quadratic"?

The word "quadratic" comes from "quad," which means "square." The highest power of x in the equation is 2, hence the name.

3. What do they tell us?

Quadratic equations help us solve problems where things change in a non-linear way, like the ball's trajectory. Solving the equation means finding the values of x that make the equation true. These values are called roots or solutions.

4. How do we solve them?

There are different ways to solve quadratic equations:

• Factoring: This involves rewriting the equation as a product of two simpler expressions.
• Quadratic Formula: This is a powerful formula that can solve any quadratic equation. It looks scary, but it's just a recipe for finding the roots.
• Graphing: You can plot the equation on a graph and see where it crosses the x-axis. Those points are the solutions.

5. Why are they important?

Quadratic equations have applications in various fields:

• Physics: Calculating the motion of objects, like our ball example.
• Engineering: Designing bridges, buildings, and other structures.
• Finance: Predicting market trends and making investment decisions.
• Computer Science: Used in algorithms and computer graphics.

6. Let's see an example:

Imagine the equation describing the ball's height is: h = -5t² + 20t + 10. Here, h is the height and t is the time. Solving this equation will tell us when the ball hits the ground (h = 0). We can use the quadratic formula or graph the equation to find the solutions.

This is just a basic introduction to quadratic equations. There's much more to learn, but hopefully, this gives you a starting point to explore further!

Solve the following quadratic equation:

x² - 5x + 6 = 0

Remember, the solutions are the values of x that make the equation true. You can use any method you like, such as factoring, the quadratic formula, or graphing. Good luck!

Student: x=0 and x=10

Unfortunately, x = 0 and x = 10 are not solutions to the equation x² - 5x + 6 = 0.

Let's check:

• For x = 0: (0)² - 5(0) + 6 = 6 ≠ 0
• For x = 10: (10)² - 5(10) + 6 = 56 ≠ 0

As you can see, neither value makes the equation true.

Try using another method like factoring or the quadratic formula to find the correct solutions.

I give you a detailed method to solve it

Solving x² - 5x + 6 = 0: A Step-by-Step Guide

Let's solve this equation using two different methods: factoring and the quadratic formula.

Method 1: Factoring

1. Identify the factors of the constant term (c) whose sum is the coefficient of the linear term (b).

   • In this case, c = 6 and b = -5.

   • We need to find two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.

2. Rewrite the middle term using the identified factors.

   • x² - 5x + 6 becomes x² - 2x - 3x + 6.

3. Group the terms and factor by grouping.

   • (x² - 2x) + (-3x + 6)

   • x(x - 2) - 3(x - 2)

4. Factor out the common binomial.

   • (x - 2)(x - 3)

5. Set each factor equal to zero and solve for x.

   • x - 2 = 0 --> x = 2

   • x - 3 = 0 --> x = 3

Therefore, the solutions to the equation are x = 2 and x = 3.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that can solve any quadratic equation. It states:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = 1, b = -5, and c = 6. Plugging these values into the formula:

x = (5 ± √((-5)² - 4(1)(6))) / (2(1))
x = (5 ± √(25 - 24)) / 2

x = (5 ± 1) / 2

This gives us two solutions:

• x = (5 + 1) / 2 = 3

• x = (5 - 1) / 2 = 2

Again, the solutions are x = 2 and x = 3.

Both methods lead to the same solutions. You can choose whichever method you find more comfortable or efficient for the specific problem you're solving.