Question

4x2+4bx-(a2-b2)=0 solved

Answer 1

Hey there!

Expand this given equation to solve it via quadratic formula;

$\bf{4x^2 + 4bx - (a^2 - b^2) = 0}$

$\bf{4x^2 + 4bx - a^2 - b^2 = 0}$

Solving it by applying quadratic formula for equation "$\bf{4x^2 + 4bx - a^2 - b^2 = 0}$"

For a required quadratic equation of

$\bf{ax^2 + bx + c = 0}$

this, for which can be represented by the quadratic formula of :

$\boxed{\bf{x_{1, \: 2} = \dfrac{- b +- \sqrt{b^2 - 4ac}}{2a}}}$

Here, a = 4,  b = 4b,  c = - a^2 + b^2.

Solving for positive and negative values respectively:

$\bf{\dfrac{- 4b + \sqrt{(4b)^2 - 16 (b^2 - a^2)}}{2 \times 4}}$

$\bf{\dfrac{- 4b + \sqrt{16b^2 - 16 (b^2 - a^2)}}{8}}$

$\bf{\dfrac{- 4b + \sqrt{16(b^2 - (b^2 - a^2))}}{8}}$

$\bf{\dfrac{- 4b + \sqrt{16a^2}}{8}}$

$\bf{\dfrac{- 4b + 4a}{8}}$

$\bf{\dfrac{4(a - b)}{8}}$

$\bf{\underline{\therefore \quad x_1 = \dfrac{a - b}{2}}}$

Similarly find the other value of "x":

$\bf{\dfrac{- 4b - \sqrt{(4b)^2 - 16 (b^2 - a^2)}}{2 \times 4}}$

$\bf{\dfrac{- 4b - \sqrt{16b^2 - 16 (b^2 - a^2)}}{8}}$

$\bf{\dfrac{- 4b - \sqrt{16(b^2 - (b^2 - a^2))}}{8}}$

$\bf{\dfrac{- 4b - \sqrt{16a^2}}{8}}$

$\bf{\dfrac{- 4b - 4a}{8}}$

$\bf{- \dfrac{4(b + a)}{8}}$

$\bf{\underline{\therefore \quad x_2 = - \dfrac{b + a}{2}}}$

Therefore the final solutions for this quadratic equations are:

$\boxed{\bf{\underline{x_1 = \dfrac{a - b}{2}}}}$

$\boxed{\bf{\underline{x_2 = - \dfrac{b + a}{2}}}}$

Hope this extremely detailed process helps you and clears the doubts for solving it via quadratic equations!!!!!!