The standard form of a quadratic equation is: **ax ^{2} + bx + c = 0.**

The roots of this quadratic are:

x =

x =

in the roots,

Depending on this discriminant, b^{2} – 4ac, the roots of a quadratic equation are:

1. *Real and distinct,* if **b ^{2} – 4ac > 0**

2.

3.

*Imp Tip: *

*Each of the two roots is x = - (^{b}/_{2a}), when b^{2} – 4ac = 0*

(i.e. when the two roots of a quadratic equation are real and equal)

We will not consider the 3

**Examples:
Examine the nature of roots of the quadratic equations:
1. x**

To examine the nature of roots of a quadratic is to find if the roots are:

For this, the

In the quadratic, a = 1, b = -12, c = 36.

b

Since the discriminant

The roots of the quadratic are

So, x = 6 is the only root of the quadratic equation

**2. x ^{2} + x + 1 = 0**

Let us use the discriminant:

b

since b

**3. x ^{2} - 5x + 6 = 0**

b

To find the two roots of the quadratic equation:

x =

x =

**4. Find p so that the roots of the quadratic equation: 9x2 – 24x + p = 0 are real and different.
Solution:** in the quadratic:

For roots to be real and distinct:

(-24)

576 – 36p > 0,

576 > 36p,

36p < 576, p < 16

If

If

Equal, if

*Note: So, when it is b^{2} – 4ac = 0, the roots are:
Real and Equal when b^{2} - 4ac = 0, or
Real and distinct when b^{2} - 4ac > 0*

**5. For what value of k, the roots of the quadratic equation: kx(x – 2) + 6 = 0 are real and equal.
Solution:** Simplify the quadratic as:

kx(x – 2) + 6 = 0 = kx

Now,

Roots of the quadratic are real and equal if

(-2k)

Either

So,

Call the two roots in the quadratic equation ax^{2} + bx + c = 0 as “p” and “q”

*Then, sum of the roots: p + q = - (^{b}/_{a}) and
product of the roots: pq = ^{c}/_{a}*

**Examples: What is the sum and product of the roots in the quadratic equations?
1. x ^{2} – 12x + 36 = 0**

Solution: in the quadratic equation,

sum of the roots =

Product of the roots =

Solution: collect x terms and constant terms at one place.

9x

9x

9x

In the standard form now,

Sum of the roots =

Product of roots =

**3. Find a so that sum and product of roots of the quadratic equation: **

**ax ^{2} + 2x + 3a = 0 **are equal.

Solution: in the quadratic

Sum of roots

product of roots

as sum and product of roots are equal,

The quadratic equation whose roots are “p” and “q” is:

**Examples:
**

**1. write the quadratic equation whose roots are 4 and 6.
Solution:** sum = 10 and product = 24

So the quadratic equation is

x

Consider the quadratic equation **ax ^{2} + bx + c = 0**

**1. Both roots are positive:**

If a and b are opposite in sign and a and c are same in sign.

**2. Both roots are Negative:**

If a, b and c are all of same sign

**3. Roots are of opposite signs:**

If a and c are of opposite signs.

**4. Roots equal but opposite in signs.**

**If b = 0**