Question

Find the roots of the following quadratic equations by factorisation:
4x2 — 4a2x + a4 — b4 = 0

Answer 1

Answer:

refer to the attachment for the solution.

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formula used: a² - b² = (a + b)(a - b)

Answer 2

Question:

Find the roots of the following quadratic equation by factorisation : 4x² - 4a²x + a² - b² = 0

Answer:

x = (a² + b²)/2 , (a² - b²)/2

Note:

• The possible values of unknown (variable) for which the equation is satisfied are called its solutions or roots .

• If x = a is a solution of any equation in x , then it must satisfy the given equation otherwise it's not a solution (root) of the equation.

Solution:

Here,

The given quadratic equation is :

4x² - 4a²x + a⁴ - b⁴ = 0 .

Now ,

Splitting the middle term of the given quadratic equation, we have,

=> 4x² - 4a²x + a⁴ - b⁴ = 0

=> 4x² - 2•2a²x + a⁴ - b⁴ = 0

=> 4x² - 2•(a² + a²)x + (a⁴ - b⁴) = 0

=> 4x² - 2•(a² + b² + a² - b²)x + (a² - b²)(a² + b²) = 0

=> 4x² - 2•[(a² + b²) + (a² - b²)]x + (a²-b²)(a²+b²) = 0

=> 4x² - 2•(a²+b²)x - 2•(a²-b²)x + (a²-b²)(a²+b²) = 0

=> 2x•[2x - (a² + b²)] - (a² - b²)•[2x - (a² + b²)] = 0

=> [2x - (a² + b²)]•[2x - (a² - b²)] = 0

Case1

=> 2x - (a² + b²) = 0

=> 2x = (a² + b²)

=> x = (a² + b²)/2

Case2

=> 2x - (a² - b²) = 0

=> 2x = (a² - b²)

=> x = (a² - b²)/2

Hence,

The roots of the given quadratic equation are :

x = (a² + b²)/2 , (a² - b²)/2