CLASS X MATHEMATICS
CHAPTER 4
QUADRATIC EQUATION
Definition of a Quadratic Equation:
A quadratic equation is a polynomial equation of degree two, which can be expressed in the standard form:
ax2 +bx+c=0
where:
a , b , and c are constants (with a ≠ 0 ),
x represents the variable or unknown.
The term ax2 is the quadratic term, bx is the linear term, and c is the constant term.
Properties of Quadratic Equations:
Quadratic equations have several important properties:
Roots or Solutions: The solutions to a quadratic equation are known as the roots. These can be real or complex
numbers, depending on the discriminant.
Discriminant: The discriminant D is given by the formula D = b2 − 4ac . It determines the nature of the roots:
If D > 0 , there are two distinct real roots.
If D = 0 , there is exactly one real root (a repeated root).
If D < 0 , there are two complex roots.
Methods of solving Quadratic Equations
There are several methods to solve quadratic equations, each suitable for different scenarios:
Factoring: This method involves expressing the quadratic equation in factored form. For example, if we have x2
+5x+6=0 , we can factor it as (x+2)(x+3)=0 . Setting each factor to zero gives the solutions x = −2 and x = −3 .
Completing the Square: This method involves rearranging the equation into a perfect square trinomial. For example,
to solve x2 + 6x +5 = 0 :
Move the constant to the other side: x2+6x=−5 .
6 2
Add ( ) = 9 to both sides: x2 +6x + 9 = 4 .
2
Factor the left side: (𝑥 + 3)2 = 4 .
Take the square root of both sides: x + 3= ± 2 .
Solve for x : x = −1 or x =−5 .
Quadratic Formula: The quadratic formula is a universal method that can solve any quadratic equation. It is given by:
̶ b ± √𝑏 2 − 4𝑎𝑐
x=
2𝑎
Using the quadratic formula, one can find the roots directly by substituting the values of a , b , and c . For example ,
to find roots of equation x2 +3x – 10 = 0
Compare the given equation with general form of quadratic equation ax2 +bx + c =0
We get a = 1 , b = 3 , c = - 10 .
Putting these values in Quadratic formula
̶ 3 ± √32 − 4×1×(−10) −3 ±√9+40 −3 ±7
x= = =
2×1 2 2
−3+7 −3−7
x= or x =
2 2
Therefore x = 2 or x = -5
Method of forming a quadratic equation from its roots:
A quadratic equation whose roots are α & β can be given by –
2
𝑥 + (𝛼 + 𝛽)𝑥 + 𝛼𝛽 = 0 or x2 − (sum of roots) x + (product of roots) = 0
For example : to find a quadratic equation whose roots are -7 and 5 .
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CLASS X MATHEMATICS
Given α = -7 and β = 5
On putting the value in formula , quadratic equation will be
𝑥 2 + (−7 + 5 )𝑥 + (−7 × 5) = 0
x 2 − 2x − 35 = 0
Practice Questions:
1. Find the roots of the quadratic equations .
(i) 2x2 - 7x + 3 = 0 (ii) 2x2 + x - 4 = 0 (iii) 4x2 + 4√3x + 3 = 0 (iv) 2x2 + x + 4 = 0
2. Find the roots of the quadratic equations by applying the quadratic formula.
(i) √2x2 + 7x + 5√2 = 0 (ii) √3x2 + 10x - 8√3 = 0 (iii) 3x2 - 2√6𝑥 + 2= 0 (iv) √3x2 - 2√2x - 2√3 = 0
3. Write the nature of roots of quadratic equation.
(i) 4x2 -12x - 9 = 0 (ii) 4x2 + 4√3 x + 3 = 0 (iii) 2 − 3√5𝑥 + 10 = 0
4. If the sum of two natural numbers is 8 and their product is 15, find the numbers.
5. Find that non–zero value of 𝑘, for which the quadratic equation 𝑘𝑥2 + 1 − 2(𝑘 − 1) + 𝑥2 = 0 has equal roots. Hence
find the roots of the equation.
6. For what value of 𝑘, are the roots of the quadratic equation (𝑥 − 2)+ 6 = 0 equal?
7. The total cost of a certain length of a piece of cloth is ₹ 200. If the piece was 5 𝑚 longer and each metre of cloths
costs ₹ 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its
original rate per metre?
8. Anil takes 6 days less than the time taken by Varun to finish a piece of work. If both Anil and Varun together can
finish that work in 4 days, find the time taken by Varun to finish the work independently.
9. Sum of the areas of two squares is 400 𝑐𝑚2 . If the difference of their perimeters is 16 𝑐𝑚, find the sides of the
two squares.
10. A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits
interchange their places. Find the number.
11. Form the quadratic equation whose roots are –
(1) 3 and 5 (2) -12 and -6 (3) 5 and -8 (4) 0 and -2 (5) ( 2 + √3 ) and ( 2 - √3 )
12. Form the quadratic equation whose sum of roots and product of roots are given below –
(1) -14 and 48 (2) 25 and -658 (3) -35 and -81 (4) 0 and -36 (5) √7 and 2√3
13. A thief runs with a uniform speed of 100 m/ minute. After one minute, a policeman runs after the thief to catch
him. He goes with a speed of 100 m/minute in the first minute and increases his speed by 10 m/minute every
succeeding minute. After how many minutes the policeman will catch the thief?
14. A person on tour has ₹ 360 for his expenses. If he extends his tour for 4 days, he has to cut down his daily
expenses by ₹ 3. Find the original duration of the tour.
15. The hypotenuse of a right triangle is 1 m less than twice the shortest side. If the third side is 1 m more than the
shortest side, find the sides of the triangle.
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