# Class 9 – Quadratic Equations

Take practice tests in Quadratic Equations

## Online Tests

Topic Sub Topic Online Practice Test
• Equations reducible to quadratic form
• Nature of roots
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• Symmetric functions of roots
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## Study Material

1. Factorization Method:

The following procedure describes the method.

Step1: Let the quadratic equation be .

Step2: Express the quadratic polynomial as the product of two linear

factors, say (px+ q), (rx + s), where p, q, r, s
(This can be done by writing b (coefficient of x)as sum of two quantities whose product is equal to ac )

Now for finding the values of x, we need to use, the zero-product rule: If a.b =0, then
a = 0 or b = 0

Step3: The given equation now becomes (px +q) (rx + s) = 0

The roots of the given quadratic equation are ,

Example: Find the roots of by factorisation method

Solution:

2. Completing the square method: Let us consideran equation .

If we want to factorize the left hand side of the equation using the method of splitting the middle term, we must determine two integer factors of 4 whose sum is 8. But,the factors of 4 are 1, 4; -1, -4; -2, -2 and 2, 2. In these cases, the sum is not 8. Therefore, using factorization, we cannot solve the given equation . Here we shall discuss a method to solve such quadratic equations.

Example:Solve:.
Solution:

Example: Solve
Solution:
We have,

Derivinga formula to find the solution of a quadratic equation:

To solve this equation we have to convert the terms containing x into a perfect square, i.e. we have to convert into a perfect square as these are the terms containing x.

To make it a perfect square add on both sides

The quadratic equation has got two roots given by

Let and be the 2 unique solutions i.e. roots of the quadratic equation. Thus we have,

Solution:On Comparison with the standard quadratic equation we find that

Theroots are 3,.

Sum of the roots: Let are roots of a quadratic equation , then

i.e.,Sum of roots

Product of the Roots: Let are roots of a quadratic equation , then

i.e.,Product of roots

Finding the quadratic equation whose roots are given:

Let are roots of a quadratic equation such that

If are roots of a quadratic equation, then the quadratic equation is.

Example: Find the quadratic equation whose rootsare –3 and 5.

Solution: Let

1. Solve by factorization method.

Solution:

2. Solve by factorization method.

Solution:

3. Solve

Solution:

Here

4. Find the sum and product of roots of .

Solution:

5. Find the quadratic eqution whose roots are 3 and .

Solution: Let

Let a and b are the two roots of the quadratic equation . Thus we have,

Here, ,then

is a perfect square.