Quadratic Equations - Formulas, Methods, and Examples

Lesson Overview

• What Is a Quadratic Equation?
• Quadratic Equations Formula
• Methods to Solve Quadratic Equations
• Quadratic Equation Graphs
• Differences Between Linear and Quadratic Equations
• Solved Examples Of Quadratic Equation

Quadratic equations are a type of equation where the highest power of the variable is 2. They are used to model many real-world situations that involve curves, such as the path of a projectile or the shape of a parabola. 

These equations help us understand how objects move under the influence of gravity, how populations grow over time, and even how sound waves travel. Quadratic equations are a fundamental part of algebra and have numerous applications in fields like physics, engineering, and computer science.

What Is a Quadratic Equation?

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The highest power of the variable is 2. Also, Quadratic equations represent parabolas when graphed.

The standard form of a quadratic equation is: 

ax² + bx + c = 0 

where:

• 'x' is the variable
• 'a', 'b', and 'c' are coefficients, which are real numbers
• 'a' ≠ 0 (If 'a' is 0, the equation becomes a linear equation)

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Quadratic Equations Practice Test Questions And Answers - Quiz

Examples of Quadratic Equations:

• 2x² + 5x - 3 = 0
• x² - 4 = 0
• -3x² + x + 10 = 0
  
  

Quadratic Equations Formula

The quadratic formula provides a direct method to find the values of 'x' or the root that satisfy the equation ax² + bx + c = 0.

The Formula:

x = [-b ± √(b² - 4ac)] / 2a

where:

• 'a', 'b', and 'c' are the coefficients from the quadratic equation.
  
  
• How to Use the formula to find values of x or the root:

1. Find the Numbers: Look at your quadratic equation. It should look like this: ax² + bx + c = 0. Find the numbers in front of x² , x, and the number without any 'x'. These are 'a', 'b', and 'c'.
   
2. Plug in the Numbers: Put the values of 'a', 'b', and 'c' into the spaces in the quadratic formula:
   x = [-b ± √(b² - 4ac)] / 2a
   
3. Calculate: Follow the order of operations (PEMDAS/BODMAS) to work out the answer. Remember that the ± symbol means you'll get two answers, one when you add and one when you subtract.

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quadratic formula Quiz questions

Example:

Solve the equation: 2x² - 5x - 3 = 0

• Identify a, b, and c.
  
  
• Substitute the values of a, b, and c in the formula.
  
• Simplify the equation to find x along with evaluating the square root.

• Separate into two equations to find both values of x.

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Quadratic Equations MCQ Test: Quiz!

Methods to Solve Quadratic Equations

Apart from solving quadratic equations with the help of the quadratic formula, there are two other methods for solving quadratic equations.

Factoring of Quadratic Equations

This method can be used when the equation can be easily factored. For this we need to rewrite the equation into a product of two simpler expressions or factors.

Example:

 Solve the equation:  x² - 5x + 6 = 0

• Factorize: Find two numbers that add up to -5 and multiply to 6. Those numbers are -2 and -3. Then find the common factor.
  
  • Set each factor to zero.
    
  • Solve for x

Completing the Square

This method is done when factoring is difficult and the coefficient of x² is 1. To do this, manipulate the equation to create a perfect square trinomial.

Example: x² + 6x - 7 = 0

• Move the constant term, here which is 7.
  
• Complete the square: Take half of the coefficient of x, square it, and add it to both sides.
• Factor and simplify
  
• Take the square root.
  
• Solve for x.

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Quiz on Quadratic Equations for Class 10 Chapter 4

Quadratic Equation Graphs

Graphing quadratic functions helps us understand their behavior visually. A quadratic function, written as f(x)ax² + bx + c = 0, forms a U-shaped curve called a parabola. The shape and direction of the parabola depend on the coefficient "a," where a ≠ 0, and a, b, and c are real numbers. 

In the vertex form, a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) is the vertex, or the highest or lowest point of the parabola. The sign of "a" determines if the parabola opens upwards (when a > 0) or downwards (when a < 0).

Here's how to graph a quadratic function step by step.

• Consider the general quadratic function f(x) = ax² + bx + c. 
  
• By completing the square, we rewrite it as f(x) = a(x + b/2a)²- D/4a, where D (the discriminant) is given by D = b² - 4ac. The vertex of the parabola is at (-b/2a, -D/4a). To graph f(x), follow these transformations:
  
• x² to ax²: Scale the parabola vertically. If "a" is negative, the parabola flips downward. The scaling depends on the size of "a."
  
  
  
• ax² to a(x + b/2a)²: Shift the parabola horizontally by |b/2a| units. The direction depends on the sign of b/2a, and the vertex moves to (-b/2a, 0).
  
• a(x + b/2a)² to a(x + b/2a)² - D/4a: Shift the parabola vertically by |D/4a| units. The direction depends on the sign of D/4a, placing the vertex at (-b/2a, -D/4a).

Graphing Quadratic Functions in Standard Form

To graph a quadratic function, you can either convert it to vertex form or use the standard form to find the axis of symmetry and y-intercept.

For example

Graph the quadratic function f(x) = 2x² + 4x + 4. Since the coefficient "a" is positive, the graph opens upwards.

The value of "a" also controls how steep the graph is; a larger "a" makes the graph thinner. To plot this function, we first find the axis of symmetry and the y-intercept.

Steps for Graphing:

• Identify the coefficient "a": For f(x) = 2x² + 4x + 4, a = 2, which means the graph opens upwards.
• Find the axis of symmetry: Use the formula x = -b/2a. Here, x = -4/(2 x 2) = -1.
• Determine the vertex: The x-coordinate of the vertex is -1 (from the axis of symmetry). To find the y-coordinate, substitute x = -1 into f(x) = 2x² + 4x + 4. This gives the vertex as (-1, 2).
• Find the y-intercept: The y-intercept is given by (0, c). For f(x) = 2x² + 4x + 4, the y-intercept is (0, 4).
• Plot the graph: Use the axis of symmetry, vertex, and y-intercept to sketch the graph of f(x) = 2x² + 4x + 4.

Differences Between Linear and Quadratic Equations

Linear equations and quadratic equations are fundamental concepts in algebra, each with distinct characteristics. Here's how they differ - 

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Principles of Factoring Quadratic Equations

Solved Examples Of Quadratic Equation

1. Solve: x² - 7x + 10 = 0

• Find two numbers that add up to -7 and multiply to 10. These numbers are -2 and -5. (x - 2)(x - 5) = 0
• Set each factor to zero: x - 2 = 0 or x - 5 = 0
• Solve for x: x = 2 or x = 5
  

2. Solve: x² + 6x + 5 = 0

• Move the constant term: x² + 6x = -5
• Complete the square: Take half of the coefficient of x (6/2 = 3), square it (3² = 9), and add it to both sides: x² + 6x + 9 = -5 + 9
• Factor and simplify: (x + 3)² = 4
• Take the square root: x + 3 = ±2
• Solve for x: x + 3 = 2 or x + 3 = -2 x = -1 or x = -5
  

3. Solve: 2x² - 3x - 2 = 0

• Identify a, b, and c: a = 2, b = -3, c = -2
• Plug in: x = [3 ± √((-3)² - 4 * 2 * -2)] / (2 * 2)
• Simplify: x = [3 ± √(9 + 16)] / 4 x = [3 ± √25] / 4 x = (3 ± 5) / 4 x = 8/4 or x = -2/4 x = 2 or x = -1/2
  

4. Solve: -x² + 4x - 3 = 0

• Multiply by -1 (optional): To make the leading coefficient positive, multiply both sides by -1: x² - 4x + 3 = 0
• Factor: (x - 3)(x - 1) = 0
• Set each factor to zero: x - 3 = 0 or x - 1 = 0
• Solve for x: x = 3 or x = 1