Quadratic Equations Notes For Class 10 Chapter 4 Download PDF

Chapter 4: Quadratic Equations Notes Download PDF

The polynomial of degree two is called quadratic polynomial and equation corresponding to a quadratic polynomial P(x) is called a quadratic equation in variable x.

Thus, P(x) = ax^{2} + bx + c =0, a ≠ 0, a, b, c ∈ R is known as the standard form of quadratic equation.

**There are two types of quadratic equation.**

(i) Complete quadratic equation : The equation ax^{2} + bx + c 0 where a ≠ 0, b ≠ 0,c ≠ 0

(ii) Pure quadratic equation : An equation in the form of ax^{2} = 0, a ≠ 0, b = 0, c = 0

The value of x for which the polynomial becomes zero is called zero of a polynomial For instance, 1 is zero of the polynomial x^{2} — 2x + 1 because it become zero at x = 1.

A real number x is called a root of the quadratic equation ax2 + bx + c =0, a 0 if aα^{2} + bα + c =0.In this case, we say x = α is a solution of the quadratic equation.

1. The zeroes of the quadratic polynomial ax^{2} + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

2. Roots of quadratic equation ax^{2} + bx + c =0 can be found by factorizing it into two linear factors and equating each factor to zero.

By adding and subtracting a suitable constant, we club the x^{2} and x terms in the quadratic equation so that they become complete square, and solve for x.

In fact, we can convert any quadratic equation to the form (x + a)^{2 }— b^{2} = 0 and then we can easily find its roots.

The expression b^{2} — 4ac is called the discriminant of the quadratic equation.

Let quadratic equation is ax^{2 }+ bx + c = 0

**Step 1.**

Find D = b^{2} — M4ac.

**Step 2.**

(i) If D > 0, roots are given by x = -b + √D / 2a , -b – √D / 2a

(ii) If D = 0 equation has equal roots and root is given by x = -b / 2a.

(iii) If D < 0, equation has no real roots.

Let the quadratic equation be ax^{2} + bx + c = 0 (a ≠ 0).

Thus, if b^{2} — 4ac ≥ 0, then the roots of the quadratic —b ± √b^{2} — 4ac / 2a equation are given by

—b ± √b^{2 }— 4ac / 2a is known as the quadratic formula which is useful for finding the roots of a quadratic equation.

(i) If b^{2} — 4ac > 0, then the roots are real and distinct.

(ii) If b^{2} — 4ac = 0, the roots are real and equal or coincident.

(iii) If b^{2 }— 4ac <0, the roots are not real (imaginary roots)

If α and β are two roots of equation then the required quadratic equation can be formed as x^{2} — (α + β)x + αβ =0

Let α and β be two roots of the quadratic equation (ax^{2 }+ bx + c = 0 then Sum of Roots: – the coefficient of x / the coefficient t of x^{2 }⇒ α + β = – b / a

αβ = constant term / the coefficient t of x^{2 }⇒ αβ = c / a

**Step 1:-**

Translating the word problem into Mathematics form (symbolic form) according to the given condition

**Step 2 :**- Form the word problem into Quadratic equations and solve them.

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- Chapter 7 : Coordinate Geometry
- Chapter 1 Real Numbers
- Chapter 2 Polynomials
- Chapter 3: Pair of Linear Equations in Two Variables
- Chapter 6 Lines and Angles
- hapter 14 : STATISTICS
- Chapter 6 : Triangles
- Chapter 8 : Introduction to trigonometry
- Chapter 4: Quadratic Equations
- Chapter 5 : Arithmetic Progressions
- Chapter 9 : Some Applications of Trigonometry
- Chapter 10 : Circles
- Chapter 11 : Constructions
- Chapter 12 : Area Related to Circles
- Chapter 13 : Surface Areas and Volumes
- Chapter 15 : Probability

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