Quadratic Equations

An equation of the form in which 2 is the highest power of x and a,b,c are any three numbers free from x is called an equation of second degree or a quadratic equation in x. Here a, b, c are constant terms; a is called the quadratic coefficient, b is called the linear coefficient and c is called the constant or free term. If a=0, then the equation is linear instead of quadratic.

Solving Quadratic Equation

A quadratic equation may be solved either by factorizing the left side( when the right side is zero) or by completing a square on the left side.

Example 1: Solve

Solution:

Therefore, either or,

Hence,

Example 2: Solve

Solution: 

Formula to Solve a Quadratic Equation:

The roots of the quadratic equation  is given by the following formula:

This formula is known as Sridhar Acharya’s formula.

Example3: Solve

Solution: According to Sridhar Acharya’s Formula, here, a=2, b=(-10), c=13

which are complex roots. To know more about complex numbers and ‘i’ refer Complex Numbers.

Equations Reducible to Quadratic Form

Many equations does not look like quadratic equations but can be reduced to quadratic form very easily. Let us see some examples.

Example 4: Solve

Solution: 

or,

or, …(1)

Now, let Then,

from (1) we get,

or, 

or,

or,

or,

or

If, , then, , which is not possible.

If, , then

or,

 

Example 5: Solve

Solution: Let us put , then

or,

or, 

If

If

Answer:

Problems Leading to Quadratic Equations

Example 7: The sum of the squares of two numbers is 233 ad one of the numbers is 3 less than twice the other. Find the numbers.

Solution: Let one of the numbers be taken as x.

the other number=(2x-3)

By the problem,

or,

If , then the other number is

If then, the other number is

Answer: The required numbers are either or, .

Sum and Product of Roots of a Quadratic Equation

If   be the roots of the quadratic equation  then, and

From these two relations we obtain the following results:

1. If the two roots  be reciprocal to each other, then,
2. If the two roots be equal in magnitude and opposite in sign then

Example 8: If the roots of the equation be in the ration 2:3, prove that .

Solution: Let the roots of  be  and .

…(1)

and, …(2)

From (1), or,

From (2),

or, (Proved)

Example 9: If the roots of the equation are denoted by and and , find the value of p.

Solution: and

Now, (given)

or,

or,

or,

Nature of Roots of a Quadratic Equation

The nature of the roots of a quadratic equation is determined by which is known as the discriminant of the quadratic equation.

• Case 1: If D is positive, then the roots are real and unequal.
• Case 2: If D is a perfect sqaure and a,b,c are all rational numbers, then the two roots are real, rational and unequal.
• Case 3: If D is positive, but not a perfect square, then is real and irrational. In this case the roots are real, irrational and unequal.
• Case 4: If D=0,  then the two roots are real and equal.
• Case 5: If D is negative, then the roots are imaginary or complex. [Refer to Example 3 of this chapter]

Example 10: Prove that the equation will have equal roots if and only if, .

Solution:

which is of the form,  where

For the given equation to have equal roots we must have,

Hence,

Formation of a Quadratic Equation with Given Roots

Any quadratic equation can be written as,

Example 11: If and be tyhe roots of the equation , form the equation where roots are and

Solution:  and

Sum of the roots of the required equation

Product of the roots of the required equation

Hence the required equation is

Conjugate Roots

Surd roots and complex roots of a quadratic equation always occur in conjugate pairs.

Example 12: Find the quadratic equation with real coefficients with one root: i)  ii)

Solution: i) Since the quadratic equation with real coefficients has a root and surd roots always occur in pairs, the other root is

Sum of the roots

Product of the roots

Hence the required equation is: or,

ii)Since one root is and complex roots always occur in pairs, the other root is

Sum of the roots

Product of the roots

Hence the required equation is:

Common Roots

Example 13: Find those values of k for which the equations and have a common root.

Solution: Let be the common root of the given equations.

Then, …(1)

and …(2)

Subtracting (2) from (1) we get, or, or,

Substituting in (1) we get,

or,

or,

or,

Exercise

1. The sum of the squares of two positive numbers is 232 and one of them is 4 less than thrice the other. Find the numbers.
2. Solve by completing the square:
3. Comment on  the nature of the roots of the equation
4. Form the quadratic equation which has the roots: a) b)
5. Solve:
6. If be the roots of the equation , find the value of:
   1. 
   2. 
7. If be the roots of the equation form an equation whose roots are:
   1.  and
   2. and
8. Find the value of for which the equation will have:
   1. Equal roots
   2. Reciprocal roots
   3. Roots whose product is 9
9. If the roots of the equation be in the ratio show that,
10. Find the equation with real coefficients whose one root is
11. If the equations and have a common root, show that