Quadratic Equations

Quadratic equations are a type of polynomial equations that have the form:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable. The most distinctive feature of a quadratic equation is the presence of x^2 term. Quadratic equations can represent a wide range of real-world phenomena, including projectile motion, fields, and various optimization problems.

Components of a quadratic equation

Let's break down the general form of the quadratic equation ax^2 + bx + c = 0 :

• a is the coefficient of x^2. It determines how "wide" or "narrow" the parabola (the graph of the quadratic equation) is.
• b is the coefficient of x. It affects the horizontal position of the vertex of the parabola.
• c is the constant term. It represents the y-intercept of the parabola, where the parabola intersects the y-axis.

Solving quadratic equations

There are several ways to solve quadratic equations:

1. Factoring
2. Using the quadratic formula
3. Completing the square
4. Graphing

1. Solving by factoring

Factoring involves writing a quadratic equation as a product. If the quadratic equation can be factored, it can be solved by applying the zero product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero.

Example:

x^2 - 5x + 6 = 0

On factoring x^2 - 5x + 6 we get:

(x - 2)(x - 3) = 0

Applying the zero product property, we get:

x - 2 = 0 or x - 3 = 0

x = 2 or x = 3

2. Using the quadratic formula

The quadratic formula can solve any quadratic equation. The formula is given as:

x = (-b ± √(b^2 - 4ac)) / (2a)

This method is reliable for all types of quadratic equations. Let's demonstrate this with the equation:

2x^2 + 3x - 2 = 0

Here, a = 2, b = 3, and c = -2. Substituting these values into the quadratic formula gives:

x = (-(3) ± √((3)^2 - 4(2)(-2))) / (2(2)) x = (-3 ± √(9 + 16)) / 4 x = (-3 ± √25) / 4 x = (-3 ± 5) / 4

This leads to the following solutions:

x = (2) / 4 = 0.5 and x = (-8) / 4 = -2

3. Completing the square

This method involves transforming the quadratic equation in such a way that the left side becomes a perfect square trinomial. Let's solve the equation using this method:

x^2 + 6x + 5 = 0

First, make sure x^2 coefficient is 1. We'll set this equation aside for the time being after we move the constant term to the other side:

x^2 + 6x = -5

Add the square of half the coefficient of x to both sides, which is (6/2)^2 = 9 :

x^2 + 6x + 9 = 4

This makes a perfect square:

(x + 3)^2 = 4

Taking the square root of both sides gives:

x + 3 = ±2

Solving for x gives:

x = -1 x = -5

4. Solving by graphing

The graphical method involves plotting the quadratic equation on the coordinate plane. The points where the parabola intersects the x-axis are the solutions of the equation. Consider the equation:

y = x^2 - 4x + 3

This can be seen in a diagram like this:

The intersection points at x = 1 and x = 3 are the solutions.

Nature of roots

The roots of a quadratic equation can be classified based on the discriminant b^2 - 4ac :

• b^2 - 4ac > 0: two distinct real roots
• b^2 - 4ac = 0: exactly one real root (or one repeated root)
• b^2 - 4ac < 0: two complex roots

Example of classification of roots

Consider:

3x^2 + 2x - 1 = 0

Calculate the discriminant:

b^2 - 4ac = (2)^2 - 4(3)(-1) = 4 + 12 = 16

Since the discriminant is greater than zero, this quadratic equation has two distinct real roots.

Graphing a quadratic - parabola

The graph of the quadratic function y = ax^2 + bx + c is a parabola. Key features include:

• Vertex: The highest or lowest point. Calculated as (-b / 2a, y), where y is the function value at -b / 2a.
• Axis of symmetry: A vertical line passing through the vertex, given by x = -b / 2a.
• Directions: If a > 0, the parabola opens upward; if a < 0, it opens downward.

Example of a graph feature

Consider the quadratic function:

y = 2x^2 - 4x + 1

Find the vertex:

x-coordinate: -(-4)/(2*2) = 1 y-coordinate: 2(1)^2 - 4(1) + 1 = -1

Thus, the vertex is at (1, -1) and the parabola opens upward. Calculating the vertex helps to sketch the exact graph as given below:

Applications of quadratic equations

Quadratic equations appear in a variety of fields, such as physics, finance, and engineering. Their real-world applications include:

• Calculating the area of a plot of land based on its dimensions.
• Determining the trajectory of an object under the influence of gravity.
• Optimizing profits in business by finding the maximum or minimum of cost and revenue functions.

Example in real life scenario

Suppose you throw a ball upward, and its height h (in meters) at any time t (in seconds) follows the quadratic equation:

h = -4.9t^2 + 20t + 1.5

Find the maximum height reached by the ball:

The maximum height occurs at the vertex. Calculate the time taken to reach the vertex:

t = -20 / (2 * -4.9) ≈ 2.04 sec

Substitute back to find the maximum height:

h = -4.9(2.04)^2 + 20(2.04) + 1.5 ≈ 21.4 m

Conclusion

Quadratic equations are fundamental in algebra and are crucial to understanding higher mathematics and real-world problem-solving. Mastering solving quadratic equations, whether by factoring, completing the square, or using the quadratic formula, develops a strong mathematical foundation.

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