The word quadratic in the term Quadratic equations is derived from quadratus, a Latin word for ‘square’. Quadratics can be defined as a polynomial equation of a second degree, which means that it comprises a minimum of one term that is squared. It is also called quadratic equations. So, we define quadratic equations as equations where the variable is of the second degree. So, it is also called “Equations of degree 2”.
The solution to the quadratic equation is the value of the unknown variable x, it will satisfy the equation. So, these solutions are called roots or zeros of quadratic equations. The roots of any polynomial are a solution for the given equation. In this article, you will learn about the Introduction to Quadratic equations, formulas, Methods of Solving Quadratic Equations, solved problems on Quadratic Equation, and so on. Refer to below for an in-depth explanation.
Introduction to Quadratic Equation
A quadratic equation is a polynomial equation, the highest power attached to a variable of order 2. So the highest power of the variable attached to the polynomial equation is two, which means that at least one term in the equation exists, which is squared.
Quadratic Polynomial: It is defined as, a polynomial of the form ax^2+bx+c, where a,b and c are real numbers and a≠0, x represents an unknown variable and, the left side has all of the fancy numbers and variables, while the right side is 0, it is called a quadratic polynomial.
Quadratic Equation: When we equate a quadratic polynomial to a constant, we get a quadratic equation. Any equation in the form of p(x)=c, where p(x) is polynomial of degree 2 and C is constant, is a quadratic equation. The solutions to a quadratic equation are known as its zero, (or) roots.
The standard form of a Quadratic Equation: The standard form of a quadratic equation is ax^2+bx+c=0, where a,b and c are real numbers and a≠0.
‘a’ is the coefficient of x^2. It is called the quadratic coefficient. ‘b’ is the coefficient of x. It is called the linear coefficient. ‘c’ is the constant term.
Find various 10th Grade Math concepts similar to Quadratic Equations all arranged efficiently
Quadratic Equation Formula
The formulas for solving quadratic equations provide students with the requisite knowledge to deal with complex numerical easily. The Below-mentioned is the general quadratic equation formula. It is used to directly obtain the roots of a quadratic equation from the standard form of the equation. So, we define it as follows:
If the equation ax^2+ bx +c = 0 is a quadratic equation, then the value of x is given by,
x = [-b±√(b2-4ac)]/2a.
By substituting the values of a,b, and c, we can directly get the roots of the equation. In a square root, the quantity is called the discriminant or D. Here’s an example that will give you an understanding of what it takes while solving quadratic equations.
Example:
Solve the Equation. The equation is x2 + 2x + 1 = 0.
Solution:
Given that a=1, b=2, c=1, and
The Discriminant D is b^2 − 4ac = 2^2 − 4×1×1 = 0
Next, using the quadratic equation formula, it will be x = (−2 ± √0)/2 = −2/2
Therefore, the value of x is −1.
Quadratic Equation Examples
The quadratic equation is in the form of (ax² + bx + c = 0)
- x² –x – 9 = 0
- 5x² – 2x – 6 = 0
- 3x² + 4x + 8 = 0
- -x² +6x + 12 = 0
Examples of a Quadratic equation with an absence of ‘C’ and -a constant term.
- -x² – 9x = 0
- x² + 2x = 0
- -6x² – 3x = 0
- -5x² + x = 0
- -12x² + 13x = 0
- 11x² – 27x = 0
The Following are few examples of a quadratic equation in factored form:
-
- (x – 6)(x + 1) = 0 [after solving the obtained result is x² – 5x – 6 = 0]
- –3(x – 4)(2x + 3) = 0 [after solving the obtained result is -6x² + 15x + 36 = 0]
- (x − 5)(x + 3) = 0 [after solving the result is x² − 2x − 15 = 0]
- (x – 5)(x + 2) = 0 [ result obtained after solving is x² – 3x – 10 = 0]
- (x – 4)(x + 2) = 0 [after solving the obtained result is x² – 2x – 8 = 0]
- (2x+3)(3x – 2) = 0 [after solving the obtained result is 6x² + 5x – 6]
The examples of quadratic equation with an absence of linear co-efficient is ‘bx’.
-
- 2x² – 64 = 0
- x² – 16 = 0
- 9x² + 49 = 0
- -2x² – 4 = 0
- 4x² + 81 = 0
- -x² – 9 = 0
How to Solve Quadratic Equations?
The quadratic Equations solving methods help in deriving solutions to complex sums and equations. While solving quadratic equations we have 4 methods, 3 algebraic methods, and 1 graphical method that are beneficial. They are:
- Factorization
- Completing the square method
- Quadratic Equation Formula
- Taking the square root
Solving Quadratic Equations Examples
Problem 1:
Find the width of a rectangle of area 336 cm. sq. The length is equal to 4 times more than twice its width?
Solution:
Given that, x cm is the width of the rectangle.
The length of the rectangle is (2x + 4)cm.
We all know that the Area of a rectangle is length x Width.
After substituting the value is,
x(2x + 4) = 336
2×2 + 4x – 336 = 0
x2 + 2x – 168 = 0
x2 + 14x – 12x – 168 = 0
x(x + 14) – 12(x + 14) = 0
(x + 14)(x – 12) = 0
x = -14, x = 12
The measurement cannot be negative.
So, the width of the rectangle is x = 12 cm.
Problem 2:
The sum of a number and its reciprocal is 5 1/5. Then the required equation is y^2 + 1/y = 265.
Solution:Â
As given in the question,
The required equation is y^2 + 1/y = 265.
5y2 – 26/y + 5 = 0
y2 + 1/y + 26/5 = 0
5y2 + 26y + 5 = 0.
Let ‘y’ be the number. Then we can write y + 1/y = 5 1/5
Thus, the equation is (y2+1)/y = 26/5
y2 + 1 = 26y/5.
Hence, 5y2 + 5 – 26y = 0 is the required equation.
Problem 3:
Solve x^2 – 16 = 0. check whether the equation is correct or not.
Solution:
As given in the question, the equation is x^2 – 16 = 0.
By algebraic identity the equation is x^2 – 42 = 0.
So, the equation is (x-4) (x+4) = 0.
Hence, the x is 4 and x is -4.
Check:
Putting the values of x in the LHS of the given quadratic equation, we get
If x = 4
x^2 – 16 = (4)^2 –16 = 16 – 16 = 0
If x = -4,
then the value is x^2 – 16 = (-4)2 – 16 = 16 – 16 = 0
Thus, the given equation is correct.
FAQs on Introduction to Quadratic Equations
1. What is a Quadratic Equation?
It is defined as the polynomial equation which has the highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed as, ax² + bx + c = 0, where x is the unknown variable and a, b and c are the constant terms.
2. What are the applications of Quadratic Equation?
The applications of Quadratic Equation are many real-life word problems that can be solved using quadratic equations. While solving some word problems, a few common quadratic equation applications are included those are speed problems and Geometry area problems.
- Solve the problems related to finding the area of quadrilateral such as rectangle, parallelogram, and so on.
- Solving the Word Problems which involve Distance, speed, time, etc.,
3. What is the use of the quadratic equations?
The quadratic equations are actually used by us in our daily lives. We will be using the equations while calculating the area, determining a product profit, and formulating the speed of the object.
4. How many types of quadratic equations?
There are 3 types of Quadratic Equations:
y = ax2 + bx + c y = (ax + c)(bx + d)
y = a(x + b)2 + c
5. What makes a problem quadratic?
It describes a problem that deals with a variable multiplied by itself, which we know as squaring. Moreover, this language originates from the area of a square as its side length multiplied itself.
6. Write the quadratic equation in the form of sum and product of roots?
The α and β are the roots of a quadratic equation, so
The Sum of the roots = α+β
The Product of the roots = αβ
Therefore, the required equation is:
x2 – (α+β)x + (αβ) = 0.