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:

- If the two roots be reciprocal to each other, then,
- 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

- 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.
- Solve by completing the square:
- Comment on the nature of the roots of the equation
- Form the quadratic equation which has the roots: a) b)
- Solve:
- If be the roots of the equation , find the value of:
- If be the roots of the equation form an equation whose roots are:
- and
- and

- Find the value of for which the equation will have:
- Equal roots
- Reciprocal roots
- Roots whose product is 9

- If the roots of the equation be in the ratio show that,
- Find the equation with real coefficients whose one root is
- If the equations and have a common root, show that

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