Revision: Algebra >> Quadratic Equations Applied Mathematics ISC (Commerce) Class 11 CISCE

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

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 

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

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.

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.

The given equation is (a² + b²)x² + 2(ac + bd)x + (c² + d²) = 0

We know, D = b² – 4ac

Thus,

D = [2(ac + bd)²] – 4(a² + b²)(c² + d²)

= [4(a²c² + b²d² + 2abcd)] – 4(a² + b²)(c² + d²)

= 4[(a²c² + b²d² + 2abcd) – (a²c² + a²d² + b²c² + b²d²)]

= 4[a²c² + b²d² + 2abcd – a²c² – a²d² – b²c² – b²d²]

= 4[2abcd – b²c² – a²d²]

= –4[a²d² + b²c² – 2abcd]

= –4[ad – bc]²

But we know that ad ≠ bc

Therefore, 

(ad – bc) ≠ 0

⇒ (ad – bc)² > 0

⇒ –4(ad – bc)² < 0

⇒ D < 0

Hence, the given equation has no real roots.

D = b² – 4ac 

Endpoints:

• < or > → hollow circle (endpoint not included)

• ≤ or ≥ → solid/dark circle (included)

Direction:

• x > a: shade to the right

• x < a: shade to the left

• a < x ≤ b: between a and b, left open, right closed