Have you observed a ball thrown up and coming down in the path of a parabola (inverted U)? What do you think the path traced by the movement of the swing would be? Is it not a parabola again (U shape)? Did you know that these trajectories can be mathematically written as “Quadratic Equations?” Let us learn more about this.
Before we dwell deep into the quadratic equations let us learn about equations in general. An equation is an alpha-numeric mathematical expression separated by an = (equal) sign. By alpha-numeric we mean algebraic expressions. An equation has three parts: The left-hand side (LHS), the right-hand side (RHS), and the equal sign. In an equation, the LHS and the RHS side are divided by the = sign, which indicates that they are equal.
Examples of the equation: 4x + 5 = 0, 567y + 5675n -z = a+2.
There are various types of equations that you might come across.
Different types of equations are:
- Linear Equation
- Quadratic Equation
- Cubic Equation
- Biquadratic Equation
Quadratic Equation
A quadratic equation is a Second-Degree equation in one variable. The general form of ax2 + bx + c=0, where a, b, c are any constants and a≠0. The solutions of the quadratic equation give the values of x that satisfy the quadratic equation. They are called the roots of a quadratic equation (α, β). Thus, x2 + 5x + 6 = 0, is a quadratic equation.
Food for thought!
- What happens if a = 0?
- Quad means 4, whereas the degree of the quadratic equation is 2. Then what is 4 in the equation ax^2 + bx + c=0 to call it quadratic?
Read further and find answers to the above questions.
Important Terminologies
Let us understand some important terminologies that may crop up while dealing with quadratic equations. Understanding them plays an important role in developing skills to solve related questions. Hence, making this learning fun and interesting. Some of them are listed below:
- Roots or Zeros of Quadratic Equation
- Nature of Roots
- Discriminant
Degree of an equation: The degree of the equation is the highest power of the variable in the equation in its simplest form.
- Examples: The degree of the equation x + y =9 is 1.
- Because the highest power of both the variables x and y is 1.
- Whereas the degree of equation x2 + y3+z=0 is 3 as that is the highest power of the variable.
Solution of an equation: It is the numerical value of the variable of the equation when substituted and simplified to satisfy the equation. The number of solutions in the equation is equal to the degree of the equation.
- Example: For the x + 3 = 0, the value x = -3 is the solution since -3 + 3 = 0.
- Since the degree of x + 3 = 0 is of degree 1 it has only 1 solution.
The solution of a quadratic equation gives two roots that can be real, complex, or equal.
Roots or Zeros of Quadratic Equation
Roots of quadratic equations are the solutions obtained by solving the given equation. And, roots are the values of the unknowns at which the polynomial or the equality holds good. It is also known as the zeroes, solutions, and intercept of the equation.
Generally, two roots are obtained from the solution of any quadratic equation. And, the roots are represented by the symbols alpha (α) and beta (β).
Nature of Roots
The roots of a quadratic equation can be:
- real and distinct,
- real and equal, or
- imaginary.
Discriminant of the quadratic equation gives a clear idea about the nature of the root without solving the equation.
Discriminant
Discriminant is a number calculated using the coefficients of the variable in the quadratic equation. It helps in predicting the nature of roots without deducing them.
For a quadratic equation ax^2 + bx + c=0, the discriminant of the equation is calculated using the formula,
D = b^2-4ac
The nature of the roots of the quadratic equation is determined by the nature of the discriminant.
- When D > 0, the roots are real and distinct.
- When D = 0, the roots are equal and the equation has only one real solution.
- When D < 0, the roots are not real but are imaginary.
Exploring Quadratic Equations
Quadratic equations are can be greatly related in real life. They are the mathematical models of parabolic trajectories and serve as the equations of motion of the same.
It is believed that Babylonians were the first to solve quadratic equations. While Greek mathematician Euclid also developed some methods to solve quadratic equations, an Indian Mathematician Sridharacharya gave the formula for solving quadratic equations.
The standard form of Quadratic equations.
A quadratic equation looks like ax^2 + bx + c=0, where a, b, c are some constants and a≠0.
For Example: –
2x^2+3x+4=0
Discriminant,
D = b^2-4ac
b^2-4ac = 3^2 – 4x2x4
Therefore, D = -21,
When D < 0, the roots are not real but are imaginary.
In the given example ‘a’ cannot be equal to zero.
Why a ≠ 0?
a should not be equal to zero because if a is equal to zero then our equation reduces to a degree 1 equation and is no longer quadratic.
ax^2 + bx + c=0
If a=0, then
0×x^2+bx+c=0
bx + c =0
The equation is linear. It’s not a quadratic equation.
Let’s verify whether some equations are quadratic equations or not.
- x^2-4x+4=x^2
The equation is not quadratic because x2 gets canceled out from LHS and RHS. And the degree of the equation reduces to 1 degree.
- (x-5)2=2×2-10x+25
It is a quadratic equation since the degree of the equation is 2.
Different Methods for Solving Zeroes of a Quadratic Equation
Any quadratic equation can be solved to obtain the roots. Some methods can be used to find the solution to quadratic equations. Some of the methods are as follows:
- Factorization Method
- Completing Square Method
- Formula Method
- Graphical Method
Each method has can be an application of a real-life situation and then obtaining a solution using the methods.
Solving Quadratic Equation by Factorization method:
The process involves making the factor of the equation and solving it.
For example,
x^2-5x+6=0
Modifying the equation,
x^2-2x-3x+6=0,
(x^2-2x-3x+6) ={x(x-2) -3 (x-2)} =(x-2) (x-3)
Equating the factor with zero, we get
x=2 and x=3
Thus, x = 2 and x = 3 are roots of the quadratic equation.
Solving Quadratic Equation by Completing Square Method
Making the Coefficient of x^2 as unity.
For example:
x^2-3x-2=0
To form the square there must be 3 terms and the form should be x^2+2bx+b^2.
x^2-3x-2=x^2-2 3/2x -2 ………….……………… (Multiplying and dividing by 2)
Now,
x^2-3x-2=x^2-2 3/2x -2 = x^2-2 3/2x +9/4- 9/4 -2
Hence,
x^2-2 3/2x +9/4- 9/4 -2=0
x^2-2 3/2x +9/4= 9/4 +2
(x-3/2)^2=17/4
(x-3/2)=±( 17/4)^1/2
x=3/2 ±√174
Sridharacharya Formula
Indian Mathematician Sridharacharya is credited for this formula. The roots of quadratic equations can be easily obtained with help of the Sridharacharya formula.
In this method quadratic equation is represented in the standard form ax^2 + bx + c=0 and x=(-b±√b^2-4ac)/2a where “a” and “b” are coefficients of x^2 and x respectively.
For Example: –
5x^2-15x-10=0
Here a=5; b=-15; c=-10
Applying Sridharacharya formula,
Hence
x=(15+√1825)/10 and x=(15-√1825)/10
Graphical Method
In this method, a quadratic function is plotted on the Cartesian coordinate system.
And all those points at which the quadratic curve intersects the x-axis are referred to as Zeroes or Roots of the quadratic.
A quadratic curve is a parabola in shape.
For all real roots of the quadratic, the parabola intersects the x-axis at most 2 points, and for complex roots, the parabola does not intersect the x-axis.
For example:
x^2-5x+6=0
The intersection of the curve to the x-axis is the root of the equation.
Here x=2 and x=3 are solutions
Relationships between roots and coefficients of the quadratic equation
- The sum of the root (α + β) is equal to – b/a : where b= coefficient of x and a= coefficient of x^2.
- The product of the roots (αβ) is c/a; where a=coefficient of x^2 and c is the constant.
Forming Quadratic Equations with the help of given roots
A quadratic equation with roots α, β can be formed by the formula and simplifying
x^2-(α+β) x+(αβ) = 0;
Conclusion:
In day-to-day life, we utilize the quadratic formula to calculate areas, calculate a product’s profit, or calculate an object’s speed.
Since it models the projectile motion, it refers to a formula that generates all of the zeros in a parabola. The quadratic formula can also be used to determine the parabola’s axis of symmetry and how many real zeros are present in the quadratic equation. Hence, whenever you are playing a game of golf, tennis, badminton, basketball, or volleyball remember that you are the creator of a quadratic equation!
Problems: –
- Identify which one of them is a quadratic equation: –
- (x-4)^2=(x^2-8x+16)
- x^2-4=x^2+4
- x^2-4x-4=2×2-4x-4
- x^2=0
- Solve for x : –
- 5x^2+10x=0
- 6x^2-30x+36=0
- 5x^2=20
- How can you find the nature of roots for the given quadratic equation 9x^2 + 5x – 2 = 0, without actually solving it?
- If the perimeter of a rectangular garden is 102 m and the area of the same garden is 620 m^2. Can you find the dimensions of the garden?
- Find the sum of zeros for the given quadratic equation, x^2 + 17 = 18x.
