Revision: Algebra >> Quadratic Equations Maths English Medium Class 10 CBSE

A value of the variable which satisfies the equation is called a root (solution).

If substituting a value of x makes the polynomial zero, that value is a root.

• A number α is called a root of ax² + bx + c = 0, if aα² + bα + c = 0

An equation with one variable, in which the highest power of the variable is two, is known as a quadratic equation.

Standard Form:

 ax² + bx + c = 0,  a ≠ 0

For example :

(i) 3x² + 4x + 7 = 0

(ii) 4x² + 5x = 0 

The set of elements representing the roots of a quadratic equation is called its solution set.

If a quadratic equation contains only two terms where one is a square term and the other is the first power term of the unknown, it is called adjected quadratic equation.

For example :

(i) 4x² + 5x = 0

(ii) 7x² − 3x = 0, etc. 

If the quadratic equation contains only the square of the unknown, it is called a pure quadratic equation.

For example :

(i) x² = 4 

(ii) 3x² − 8 = 0, etc.

Those values of x which do not satisfy ax + b ≥ 0 and cx + d ≥ 0 are called extraneous values. 

For the quadratic equation ax² + bx + c = 0, a ≠ 0; the expression b² − 4ac is called the discriminant and is, in general, denoted by the letter 'D'.

Thus, discriminant D = b² − 4ac.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

If the product of two real numbers is zero, then at least one of the numbers is zero.

• That is, if

  ab = 0 ⇒ a = 0 or b = 0. 
• This rule is used to find solutions after factorisation.

Given the roots of the equation (q – r)x² + (r – p)x + (p – q) = 0 are equal.

∴ Discriminant (D) = 0

⇒ b² – 4ac = 0

⇒ (r – p)² – 4 × (q – r) × (p – q) = 0

⇒ r² + p² – 2pr – 4[qp – q² – rp + qr] = 0

⇒ r² + p² – 2pr – 4qp + 4q² + 4rp – 4qr = 0

⇒ r² + p² + 2pr – 4qp – 4qr + 4q² = 0

⇒ (p + r)² – 4q(p + r) + 4q² = 0

Let (p + r) = y

⇒ y² – 4qy + 4q² = 0

⇒ (y – 2q)² = 0

⇒ y – 2q = 0

⇒ y = 2q

⇒ p + r = 2q

Hence proved.

1. Write the given equation in the standard form

   ax² + bx + c = 0

2. Identify the values of a, b, and c.

3. Find the value of the discriminant

   D = b² − 4ac

4. Substitute the values of a, b, and D in the formula

   \[x=\frac{-b-\sqrt{b^2-4ac}}{2a}\]​

5. Simplify to obtain the roots.

D = b² – 4ac 

In the factorisation method, the quadratic expression is written as a product of two linear factors

1. Clear fractions and brackets, if any.

2. Transpose all terms to one side to get the standard form
   ax² + bx + c = 0
3. Factorise the quadratic expression into two linear factors.

4. Put each factor equal to zero (using the zero product rule).

5. Solve the resulting linear equations to obtain the roots.