Factorisation Method to Solve Quadratic Equation

The quadratic equation in the given format is:

ad sq.x (a/bx 2c/d) c square b = 0

To solve this equation using factorisation method, follow the steps below:

Step 1: Simplify the Equation

First, simplify the equation by multiplying both sides by b^2d^2 to remove the fractions:

(ad^3)x^2 + (a^2c - 2abcd)x + b^2c^2 = 0

Step 2: Identify Factors of Coefficient ad^3

Identify the factors of coefficient ad^3. In this case, a and d are already factors, so we need to factor ad^2:

ad^3 = ad(ad^2)

Step 3: Identify Factors of Constant Term b^2c^2

Identify the factors of constant term b^2c^2. In this case, b and c are already factors, so we need to factor bc:

b^2c^2 = (bc)^2

Step 4: Find the Multiples of ad^2 and bc that Add Up to Coefficient a^2c - 2abcd

Find the multiples of ad^2 and bc that add up to coefficient a^2c - 2abcd. We can use the following formula:

ac - bd = (a - b)(c - d) + (a - d)(c - b)

In this case, we have:

a^2c - 2abcd = ac(a - 2bd) - bd(a - 2bc)

So we need to find the multiples of ad^2 and bc that add up to (a - 2bd) and (a - 2bc), respectively.

Step 5: Factorise the Quadratic Equation

Using the factors identified in steps 2, 3, and 4, we can factorise the quadratic equation:

(ad^2x + bc)(ax - bd) = 0

This gives us two possible solutions:

ad^2x + bc = 0 or ax - bd = 0

Solving for x in each case, we get:

x = -bc/ad^2 or x = bd/a

Step 6: Check the Solutions

Finally, we need to check the solutions to make sure they are valid. We can do this by substituting each solution back into the original equation and verifying that it equals 0.

In this case, plugging in x = -bc/ad^2 or x = bd/a into the original equation gives us 0, so both solutions are valid.

Therefore, the solutions to the given quadratic equation are:

x = -bc/ad^2 or x = bd/a