# Mathematics - Class 10 / Grade 10

- Introduction To Quadratic Equations / Basic Of Quadratic Equations
### Introduction To Quadratic Equations / Basic Of Quadratic Equations

**Tags:**Example of a quadratic Equation for tenth Grade, Quadratic definition, Quadratic Equation Notes for Class 10, CBSE Revision Notes for CBSE Class 10 Mathematics Quadratic Equation, What is a quadratic equation? Quadratic Equation Practice Page for Grade X, Lesson on Quadratic Equation for 10^{th}Standard, Value of Quadratic Polynomial, Zero of a Quadratic Polynomial, Worksheet PDF on Quadratic EquationsQuadratic Equations

**Introduction**A polynomial of degree 2 is called quadratic polynomial. The name Quadratic comes from "quad" means square, because the variable gets squared (like x

^{2}).The general form of a quadratic polynomials is

**a**x^{2}+**b**x +**c**, where a, b, c are real numbers such that a ≠ 0 and x is a variable. When we equate this polynomial to zero, we get a quadratic equation i.e.**a**x^{2}+**b**x + c = 0 and when we write the terms of p(x) in descending order of their degrees, then we get the__"Standard Form"__of a Quadratic**a**x^{2}+**b**x + c = 0. We can also denote a quadratic polynomial**a**x^{2}+**b**x +**c**by**p(x)**i.e.**p(x)**=**a**x^{2}+**b**x + c. It is also called an "Equation of degree 2" (because of the power "2" on the**x**).The real numbers

**a**,**b**and**c**are known as the__coefficients__and are also known as the__real constants__because they are fixed and do not depend on the values of variable x.Here are some more examples:

**Example 1:**2x^{2}-3x + 4 = 0 (Here a=2, b=-3 and c=4)**Example 2:**x^{2}+ 9 = 0 (Here a=1, and we don't usually write "1x^{2}", b=0 and c=9)**Example 3:**x^{2}= 3x – 1 (In this case we will move all terms to left hand side i.e. x^{2}− 3x + 1 = 0. Here a=1, b=−3, c=1)**Example 4:**2(p^{2}– 2p) = 5 (Solve the brackets and move all terms to left hand side i.e. 2p^{2}− 4p − 5 = 0. Here a=2, b=−4, c=−5)**Example 5:**Jia and Tia together have 45 pencils. Both of them lost 5 pencils each, and the product of the number of pencils they now have is 124. Find out how many pencils each had.**Solution:**Let the number of pencils Jia had be*x*.Then the number of pencils Tia had = 45 –

*x*(left out of total belongs to Tia)The number of pencils left with Jia, when she lost 5 pencils =

*x*– 5The number of pencils left with Tia, when she lost 5 pencils

= 45 –

*x*– 5= 40 –

*x*

**Value of Quadratic Polynomial**Let’s p(x) =

**a**x^{2}+**b**x + c be a quadratic polynomial and let ‘α’ be the real number.Then,

**a**α^{2}+**b**α + c is known as the__value of the quadratic polynomial__p(x) and it is denoted by p (α), i.e., p(α)=**a**α^{2}+**b**α + c.Thus, the value of a quadratic polynomial

**a**x^{2}+**b**x + c at x = α is the value of the expression**a**x^{2}+**b**x + c obtained by substituting x = α.**Example1:**Find the values of the quadratic polynomial p(x) = 2x^{2}-3x + 5 at x = -1 and x = 2.**Solution:**Here**,**p(x)**=**2x^{2}-3x + 5.__Case 1:__Now substitute the value of x by -1 in the given quadratic polynomial.Therefore,

p (-1) = 2 x (-1)

^{2}– 3 x (-1) + 5= 2 + 3 + 5

= 10.

__Case 2:__Now substitute the value of x by 2 in the given quadratic polynomial.Therefore,

p (2) = 2 x (2)

^{2}– 3 x (2) + 5= 8 - 6 + 5

= 7.

**Zero of a Quadratic Polynomial**A real number ‘α’ is called a

__zero of a quadratic polynomial__p(x) =**a**x^{2}+**b**x + c, if p (α) = 0. Thus, a zero of a quadratic polynomial is the value of the variable for which the value of the polynomial becomes zero. Every quadratic polynomial can have__at most two zeros.__**Example 1:**Show that 2 is a zero of the quadratic polynomial p(x) = x^{2}= x – 6.**Solution:**Here, p(x) = x^{2}= x – 6.Substitute value of x by 2

Therefore, p (2) = 2

^{2}+ 2 – 6= 4 + 2 – 6

= 0

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