Revision: Algebra >> Quadratic Equations Maths (English Medium) ICSE Class 10 CISCE

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.

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.

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