Question

Solve the following quadratic equations by factorization:
(m/nx²)+(n/m)=1-2x

Answer 1

The factorisation of the given equation = ($\dfrac{m}{n}$x + 1 + $\sqrt{\dfrac{m}{n}}$)($\dfrac{m}{n}$x + 1 - $\sqrt{\dfrac{m}{n}}$) = 0

Step-by-step explanation:

The given quadratic equation:

$\dfrac{m}{n}x^2+\dfrac{n}{m}=1-2x$

Solve the given quadratic equations by factorization method.

Let a = $\dfrac{m}{n}$

The given quadratic equation becomes:

$ax^2+\dfrac{1}{a} =1-2x$

⇒ $\dfrac{a^2x^2+1}{a} =1-2x$

By crossmultiplication method,

⇒ $a^2x^2$ + 1 = a(1 - 2x)

⇒  $a^2x^2$ + 1 = a - 2ax

⇒  $a^2x^2$ + 1 + 2ax = a

⇒ $(ax + 1)^2$ - a = 0

Using the algebraic identity,

$(a+b)^{2}=a^{2}+2ab+b^{2}$

⇒ $(ax + 1)^2$ - $\sqrt{a}^2$ = 0

⇒ (ax + 1 + $\sqrt{a}$)(ax + 1 - $\sqrt{a}$) = 0

Using the algebraic identity,

$a^{2} -b^{2}$ = (a + b)(a - b)

Put a = $\dfrac{m}{n}$ , we get

($\dfrac{m}{n}$x + 1 + $\sqrt{\dfrac{m}{n}}$)($\dfrac{m}{n}$x + 1 - $\sqrt{\dfrac{m}{n}}$) = 0

∴ The factorisation of the given equation = ($\dfrac{m}{n}$x + 1 + $\sqrt{\dfrac{m}{n}}$)($\dfrac{m}{n}$x + 1 - $\sqrt{\dfrac{m}{n}}$) = 0