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# Quadratic Equations CBSE Notes Class 10 Maths Chapter 4

## Quadratic Equations

The standard form of a **quadratic polynomial** is **p(x) = ax ^{2} + bx + c**. Then,

**ax**is the

^{2}+ bx + c = 0**general equation of quadratic equations**, where a, b and c are real and a 0.

Since the degree of a quadratic equation is two, then the quadratic equation is satisfied by exactly two roots which may be real of imaginary.

### Class 10 Maths Chapter 4 Examples 1:

Check whether the equation x^{3} – 4x^{2} – x + 1 = (x – 2)^{3} is quadratic or not.

x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}

⇒ x^{3} – 4x^{2} – x + 1 = x^{3 }– 8 – 6x^{2} + 12x

⇒ – 4x^{2} – x + 1 =^{ }– 8 – 6x^{2} + 12x

⇒ 2x^{2} – 13x + 9 = 0

It is of the form ax^{2} + bx + c = 0. So, the given equation is a quadratic equation.

## Solution of Quadratic Equations : By Factorization

A real number x = α is a root of the quadratic equation ax^{2} + bx + c = 0, a ≠ 0 if α satisfies the quadratic equation it means, a α^{2} + bα + c = 0.

### Class 10 Maths Chapter 4 Examples 2:

Find the roots of the following quadratic equations by factorization: x^{2} – 3x – 10 = 0.

x^{2} – 3x – 10 = 0

⇒ x^{2} – 5x + 2x – 10 = 0

⇒ x(x – 5) + 2(x – 5) = 0

⇒ (x + 2)(x – 5) = 0

⇒ x = -2 or 5

## Solution of Quadratic Equations : By Method of Completing Square

### Class 10 Maths Chapter 4 Examples 2:

Find the roots of the following quadratic equations by method of completing the square: 3x^{2} – 5x + 2 = 0.

Given that 3x^{2} – 5x + 2 = 0

## Solution of Quadratic Equations : By Quadratic Formula

The roots of the quadratic equation **ax ^{2} + bx + c = 0** are

Thus, the sum of the roots is **α + β = -b/a** and the product of the roots is **αβ = c/a**.

### Class 10 Maths Chapter 4 Examples 3:

Solve the quadratic equation 2x^{2} + x – 528 = 0 by using quadratic formula.

Given that a = 2, b = 1 and c = -528

## Nature of Roots of Quadratic Equations

In a quadratic equation **ax ^{2} + bx + c = 0**, the value

**D = b**is known as discriminant of the quadratic equation.

^{2}– 4acThe nature of the roots of the quadratic equation depends on the discriminant.

**Case I**: When **D > 0 **

In this case, the roots **α **and** β** of the quadratic equation are **real and unequal**.

**Case II**: When **D = 0 **

In this case, the roots **α **and** β** of the quadratic equation are **real and equal**.

**Case III**: When **D < 0 **

In this case, the roots **α **and** β** of the quadratic equation are **imaginary and distinct**.

### Class 10 Maths Chapter 4 Examples 4:

Find the nature of the roots of the quadratic equation 2x^{2} – 4x + 3 = 0.

Given that a = 2, b = -4 and c = 3.

Since the discriminant b^{2} – 4ac = (– 4)^{2} – (4 × 2 × 3) = 16 – 24 = – 8 < 0

Thus, So, the given equation has no real roots.

### Class 10 Maths Chapter 4 Examples 5:

Find the values of k for each of the quadratic equation 2x^{2} + kx + 3 = 0, so that they have two

equal roots.

Given that a = 2, b = k and c = 3.

Since the given equation has equal roots, then

D = b^{2} – 4ac = 0

⇒ k^{2} – (4 × 2 × 3) = 0

⇒ k^{2} = 24

⇒ k = 2√3.

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