Quadratic Equations

------------------ Main topics to be covered ----------------------
------------ Standard Form of a Quadratic Equation ---------------
------------ Check Whether the Following are Quadratic Equations ---
------------ Represent the Following Situations in the Form of Quadratic Equations

Quadratic equations play a vital role in Class X mathematics, forming the foundation for many advanced math topics. We will help you understand the standard form of quadratic equations, how to represent situations mathematically, and techniques for solving these equations using factorization. We’ll also cover how to check if an equation is quadratic, find roots, and understand the nature of roots using the discriminant.

Standard Form of a Quadratic Equation : ax² +bx + c=0

$a$ $, b and$ $c are constants, with$ $a \neq 0$ .
x is the variable

$a x^{2}$ represents the quadratic term, bx represents the linear term, and $c$ is the constant term.

Check Whether the Following are Quadratic Equations

To check if an equation is quadratic, we look for the highest power of the variable, which should be 2. Let’s look at some examples:

Examples of quadratic equations:

2x² - 2x + 3=0, 3x² - 2x =0, x² + 4 =0

Examples of equations which are not quadratic:

x³ - 2x² - 2x+3=0	Reason: The highest power on x is 3.
x² + 2x + x^-1= 0	Reason: The power on x is -1, not a whole number.
4x² - 2 $\sqrt{x}$	Reason: The variable x is under a radical symbol.
x² + 2 /x	Reason: The variable x is in the denominator.

Represent the Following Situations in the Form of Quadratic Equations

EXERCISE 4.1 Q2(i) The area of a rectangular plot is 528 m² . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Steps	Reasoning
Let breadth = x m Length = 2x+1 m	Assign the variable to the unknown quantity The length is one more than twice its breadth.
x(2x+1)=528	Area = Length * breadth = 528
2x²+x-528=0	General form: ax²+bx+c=0

EXERCISE 4.1 Q2 (ii) The product of two consecutive positive integers is 306. We need to find the integers.

Steps	Reasoning
Let the required numbers are x and x+1	Assign the variables to the unknown quantity
x(x+1)=306	Using: Product is 306
x²+x-306=0	General form: ax²+bx+c=0

EXERCISE 4.1 Q2 (iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

Steps	Reasoning
Rohan's age= x years Mother's age = x+26 yrs	Assign the variable to the unknown quantity Rohan’s mother is 26 years older than him.
(x+3)(x+29)=360	After 3 years their ages will be x+3 and x+26+3
x²+32x-273=0	General form: ax²+bx+c=0

EXERCISE 4.1 Q2 (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Steps	Reasoning
speed of the train=x km/h	Assign the variable to the unknown quantity
time=480/x hours	time = distance/speed
time = 480/(x-8)	Reduced speed = x-8 km/h
480/(x-8) -480/x =3	Difference of time is 3 hours

Quadratic Equations

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