Quadratic Equation A quadratic equation is a second-degree polynomial equation in a single variable. It is called “quadratic” because the highest exponent of the variable is 2. In other words, any polynomial equation of degree 2 is known as a “Quadratic Equation”.
Table of Contents
Introduction of Quadratic Equation:
A quadratic equation is a polynomial equation of degree 2. It is written in the standard form:
ax2 + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.
Examples of Quadratic Equations:
- x2 + 5x + 6 = 0
- x2 + 4x – 5 = 0
- 2x2 – 9 = 0
Examples of Non-Quadratic Equations:
- x3 + 5x2 + 4x + 6 = 0 (Degree is 3, not 2)
- 4x – 5 = 0 (Degree is 1, not 2)
Methods of Solving Quadratic Equations
There are several methods to solve quadratic equations:
- Factorization Method
- Completing the Square Method
- Quadratic Formula Method
Factorization Method:
Steps:
- Convert the equation into the standard form ax2 + bx + c = 0
- Find two numbers whose sum is b and whose product is ac.
- Split the middle term using these numbers.
- Factorize and find the values of x.
Example:
x2 + 5x + 6 = 0
Solution:
- Find two numbers whose sum is 5 and the product is 6 → (2, 3)
- Rewrite: x2 + 2x + 3x + 6 = 0
- Factorize: x (x+2)+3 (x+2) = 0
- Common Factor: (x + 3)(x + 2) = 0
Hence the value of x = -3 or -2.
Completing the Square Method:
This method involves converting a quadratic equation into a perfect square trinomial.
Example: x2 + 4x – 5 = 0
Solution: x2 + 4x – 5 = 0
x2 + 2*x*2 + 22 – 22 – 5 = 0
( x + 2 )2 – 4 – 5 = 0
(x + 2)2 – 9 = 0
(x + 2)2 – 32 = 0
(x + 2 + 3)(x + 2 – 3) = 0
(x + 5)(x – 1) = 0
Hence, the value of x is -5 or 1.
Quadratic Formula Method:
The quadratic formula is derived from completing the square and can be used to find the roots of any quadratic equation:
x = {-b ± √(b2 – 4ac)}/2a
or
x = { -b ± √D }/2a
where, D = b2 – 4ac it is called Discriminant.
Roots depends on the value of D:
- D > 0 : Two distinct real roots.
- D < 0 : One repeated real root.
- D = 0 : No real roots (complex roots).
Example:
x2 + x – 182 = 0
Compare with the standard form of Quadratic Equation:
ax2 + bx + c = 0 where, a = 1, b = 1 and c = -182
Using the quadratic formula:
x = {-1 ± √(12 – 4(1)(-182))}/2(1)
x = {-1 ± √(1 + 728)}/2
x = (-1 ± √729)/2
x = (-1 ± 27)/2
Hence, the value of x is equal to 13 or -14
Nature of Roots (Discriminant):
The values of x that satisfy the quadratic equation are called roots.
The discriminant (D) of a quadratic equation is:
D = b2 – 4ac
Types of Roots Based on D:
- D > 0 → Two distinct real roots.
- D = 0 → One repeated real root.
- D < 0 → No real roots.
Roots of a Quadratic Equations
- Constant Polynomial: A polynomial of degree 0, e.g., P(x) = c.
- Linear Polynomial: A polynomial of degree 1, e.g., P(x) = ax + c.
- Quadratic Polynomial: A polynomial of degree 2, e.g., P(x) = ax2 + bx + c.
- Cubic Polynomial: A polynomial of degree 3, e.g., P(x) = ax3 + bx2 + cx + d.
Graphical Representation of Quadratic Equations
A quadratic equation represents a parabola when graphed. The shape of the parabola depends on the coefficient a:
- If a > 0: The parabola opens upwards.
- If a < 0: The parabola opens downwards.
The point where the parabola intersects the x-axis represents the roots of the equation. If the discriminant is negative, the parabola does not intersect the x-axis.
Applications of Quadratic Equations:
Quadratic equations are used in real-life situations like:
- Area problems
- Speed-time problems
- Work and time problems
Example:
The product of two consecutive positive integers is 182. Find the integers.
Solution:
Let the integers be x and x + 1.
x(x + 1) = 182
x2 + x – 182 = 0
Compare with the standard form of Quadratic Equation:
ax2 + bx + c = 0 where, a = 1, b = 1 and c = -182
Using the quadratic formula:
x = {-1 ± √(12 – 4(1)(-182))}/2(1)
x = {-1 ± √(1 + 728)}/2
x = (-1 ± √729)/2
x = (-1 ± 27)/2
Hence, the value of x is equal to 13 or -14 but the value of x is the only possible positive value due to the given in question.
Therefore, the value of x will be 13 and the required consecutive integers will be 13 and 14.
Summary
- A quadratic equation is a second-degree polynomial.
- The roots can be found using methods like factorization, completing the square, or the quadratic formula.
- The discriminant helps determine the nature of the roots.
- Quadratic equations have various real-life applications and can be represented graphically as parabolas.
Frequently Asked Questions (FAQs)
What is a quadratic equation?
ax2 + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.
What is the degree of a quadratic equation?
How do you find the degree of a polynomial?
How many solutions can a quadratic equation have?
1. Two distinct real roots,
2. One repeated real root,
3. No real roots (complex roots).
What is the discriminant in a quadratic equation?
D = b2 – 4ac
If D > 0: there are two distinct real roots.
If D < 0: there are no real roots (the roots are complex).
If D = 0: there is one real repeated root.
How do you solve a quadratic equation?
a) Factorization: Express the quadratic expression as the product of two binomials.
b) Completing the Square: Convert the quadratic into a perfect square trinomial.
c) Quadratic Formula: Use the formula x = {-b ± √(b2 – 4ac)}/2a to find the roots.
d) Graphical Method: Plot the equation on a graph and find where it intersects the x-axis (if applicable).
What is the quadratic formula?
x = {-b ± √(b2 – 4ac)}/2a
It provides the solutions for x in terms of the coefficients a, b, and c.
What are real and complex roots?
Complex roots occur when D<0, meaning the solutions involve imaginary numbers.
Can a quadratic equation have only one solution?
What is the graph of a quadratic equation?
What are the applications of quadratic equations?
calculating areas,
determining the trajectory of objects,
solving problems in physics (e.g., motion under gravity),
optimizing business models, and financial calculations.
Learn More
Subscribe “JNG ACADEMY” for more of the best content.
Important Links
YouTube Tutorials
Thank You So Much, Guys… For Visiting our Website.