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Quadratic Equations Class 10 Notes Maths Chapter 4

What is Quadratic Equation?

  • A quadratic polynomial of the form ax2 + bx + c, where a ≠ 0 and a, b, c are real numbers, is called a quadratic equation when ax2 + bx + c = 0.

Quadratic Equations Class 10 Notes Maths Chapter 4

  • Here a and b are the coefficients of x2 and x respectively and ‘c’ is a constant term.
  • Any value is a solution of a quadratic equation if and only if it satisfies the quadratic equation.

How to Find the Solution of a Quadratic Equation by Factorisation?

For a quadratic equation, a real number α is called the root of a quadratic equation ax2+bx+c =0. Hence, we can write aα2 + bα + c = 0. So, x= α is the solution of a quadratic equation or the root of a quadratic equation. In other words, α satisfies the given quadratic equation.

Note: The zeros of the quadratic equation ax2+bx+c = 0 are the same as the root of the quadratic equation ax2+bx+c = 0.

Example: Solve the quadratic equation 2x2+x-300 = 0 by the factorisation method.

Solution: 

Given quadratic equation: 2x2+x-300 = 0

By using factorisation, the quadratic equation 2x+ x -300 = 0 is written as:

2x2 – 24x+25x -300 = 0

2x(x-12) +25(x-12) =0

(i.e) (x-12)(2x+25) = 0

Therefore, x-12=0 and 2x+25 = 0

x-12 = 0

Therefore, x= 12.

Similarly, 2x+25 = 0

2x= -25

x =-25/2

x = -12.5.

Hence, the roots of the quadratic equation 2x2+x-300 = 0 are 12 and -12.5.

Question for Short Notes: Quadratic Equations
Try yourself:Which of the following is true about the roots of the quadratic equation ax2 + bx + c = 0?
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Quadratic Formula 

  • The roots, if a quadratic equation ax2 + bx + c = 0 are given by:

Quadratic Equations Class 10 Notes Maths Chapter 4

  • Here, the value b2 – 4ac is known as the discriminant and is generally denoted by D. ‘D’ helps us to determine the nature of roots for a given quadratic equation. Thus D = b2 – 4ac.

Example: Find the roots of quadratic equation x2 - 7x + 10 = 0 using quadratic formula.

Solution: 

Here, a = 1, b = -7 and c = 10. Then by quadratic formula:

Quadratic Equations Class 10 Notes Maths Chapter 4

Therefore, x = 2, x = 5.

Nature of Roots of a Quadratic Equation

  • If D = 0 ⇒ The roots are Real and Equal.
  • If D > 0 ⇒ The two roots are Real and Unequal.
  • If D < 0 ⇒ No Real roots exist.

Example: Discuss the nature of the roots of the quadratic equation 2x2 – 8x + 3 = 0.

Solution: Here the coefficients are all rational. The discriminant D of the given equation is

D = b2 – 4ac = (-8)2 – 4 x 2 x 3

= 64 – 24

= 40 > 0

Clearly, the discriminant of the given quadratic equation is positive but not a perfect square. Therefore, the roots of the given quadratic equation are real, irrational and unequal.

Relationship between roots and Coefficients of Quadratic Equation

If α and β are the roots of the quadratic equation, then Quadratic equation is 

x2 – (α + β) x + αβ = 0 

OR

 x2 – (sum of roots) x + product of roots = 0

where,

  • Sum of roots (α + β) = Quadratic Equations Class 10 Notes Maths Chapter 4
  • Product of roots (α x β) = Quadratic Equations Class 10 Notes Maths Chapter 4

Example: If α and β are the roots of the equation x2 - 4x + 2 = 0, find the value of

i) α2 + β2

ii) α2 - β2

iii) α3 - β3

iv)1/α  + 1/ β

Solution:

The given equation is x2 - 4x + 2 = 0 ...................... (i)

According to the problem, α and β are the roots of the equation (i)

Therefore,

Quadratic Equations Class 10 Notes Maths Chapter 4

(i) Now α2 + β2 = (α + β)2 - 2αβ = (4)2 – 2 x 2 = 16 – 4 = 12.

(ii) α2 - β2 = (α + β)( α - β)

Now (α - β)2 = (α + β)2 - 4αβ = (4)2 – 4 x 2 = 16 – 8 = 8

⇒ α - β = ± √8

⇒ α - β = ± 2√2

Therefore, α2 - β2 = (α + β)( α - β) = 4 x (± 2√2) = ± 8√2.

(iii) α3 + β3 = (α + β)3 - 3αβ(α + β) = (4)3 – 3 x 2 x 4 = 64 – 24 = 40.

(iv) Quadratic Equations Class 10 Notes Maths Chapter 4

The document Quadratic Equations Class 10 Notes Maths Chapter 4 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Quadratic Equations Class 10 Notes Maths Chapter 4

1. What is a quadratic equation and how is it defined?
Ans.A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). The highest exponent of the variable \( x \) is 2, which gives it the name "quadratic."
2. How can I solve a quadratic equation by factorisation?
Ans.To solve a quadratic equation by factorisation, you need to express the quadratic in the form \( (px + q)(rx + s) = 0 \). To do this, find two numbers that multiply to give \( ac \) (where \( a \) is the coefficient of \( x^2 \) and \( c \) is the constant term) and add to give \( b \) (the coefficient of \( x \)). Once factored, set each factor to zero and solve for \( x \).
3. What is the quadratic formula and when is it used?
Ans.The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It is used to find the solutions of any quadratic equation when factorisation is difficult or impossible, and it provides the values of \( x \) directly.
4. What is the nature of roots of a quadratic equation?
Ans.The nature of the roots of a quadratic equation can be determined using the discriminant \( D = b^2 - 4ac \). If \( D > 0 \), there are two distinct real roots. If \( D = 0 \), there is one real root (a repeated root). If \( D < 0 \), the roots are complex or imaginary.
5. How are the roots of a quadratic equation related to its coefficients?
Ans.The relationship between the roots (let's say \( r_1 \) and \( r_2 \)) and the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \) is given by Vieta's formulas: \( r_1 + r_2 = -\frac{b}{a} \) (sum of the roots) and \( r_1 \cdot r_2 = \frac{c}{a} \) (product of the roots). These relationships help in understanding how the coefficients affect the roots.
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