Question

Solve the equation,

$\sqrt[4]{2x-1} = \dfrac{x^2}{4}+\dfrac{3}{4}$

Answer 1

$\rm{\sqrt[4]{\rm{2 x-1}}=(x^2/4)+(3/4)}$

Take 4 as a common number denominator for both x² and 3

$\longrightarrow\rm{\sqrt[4]{\rm{2 x-1}}=(x^2+3)/4}$

Multiply both Sides of the Equation with 4

$\longrightarrow\rm{\sqrt[4]{\rm{2 x-1}}\times 4=x^2+3}$

Take Both Sides of the Equation to the power 4

$\longrightarrow\rm{\left(\sqrt[4]{\rm{2x-1}}\cdot \:4\right)^4=\left(x^2+3\right)^4}$

Expand the Left Hand Side Part of the Equation

$\longrightarrow\rm{\rm{(2x-1)}\cdot \:256=\left(x^2+3\right)^4}$

$\longrightarrow\rm{\rm{512x-256}=\left(x^2+3\right)^4}$

Expand the Right Hand Side Part of the Equation

$\longrightarrow\rm{\rm{512x-256}=\left(x^2+3\right)^4}$

$\longrightarrow\rm{512x-256=x^8+12x^6+54x^4+108x^2+81}$

Add 256 to Both Sides of the Equation

$\longrightarrow\rm{512x-256+256=x^8+12x^6+54x^4+108x^2+81+256}$

$\longrightarrow\rm{512x=x^8+12x^6+54x^4+108x^2+81+256}$

$\longrightarrow\rm{512x=x^8+12x^6+54x^4+108x^2+337}$

Subtract 512x from Both Sides of the Equation

$\longrightarrow\rm{512x-512x=x^8+12x^6+54x^4+108x^2+337-512x}$

$\longrightarrow\rm{0=x^8+12x^6+54x^4+108x^2+337-512x}$

$\longrightarrow\rm{x^8+12x^6+54x^4+108x^2+337-512x=0}$

Factor Left Hand Side of the Equation

$\longrightarrow\rm{\left(x-1\right)^2\left(x^6+2x^5+15x^4+28x^3+95x^2+162x+337\right)=0}$

Use the Zero Factor Principle : If ab = 0, then a = 0 or b = 0

$\longrightarrow\rm{x-1=0\quad \mathrm{or}\quad \:x^6+2x^5+15x^4+28x^3+95x^2+162x+337=0}$

$\longrightarrow\rm{x-1+1=0+1\quad \mathrm{or}\quad \:x^6+2x^5+15x^4+28x^3+95x^2+162x+337=0}$

$\longrightarrow\rm{x=1\quad \mathrm{or}\quad \:x^6+2x^5+15x^4+28x^3+95x^2+162x+337=0}$

$\longrightarrow\rm{x=1\quad \mathrm{or}\quad \:\mathrm{No\:Solution\:for}\:x\in \mathbb{R}}$

Therefore, The Value of x is equal to 1

Answer 2

Answer:

Answer in the attachment.