Answer :
Alright class, let's solve the equation \(\left(2x^2 + 1\right)^2 = 4(x - 2)\) step-by-step:
1. Expand both sides:
On the left-hand side, we have a squared binomial:
[tex]\[ \left(2x^2 + 1\right)^2 = \left(2x^2 + 1\right) \left(2x^2 + 1\right) \][/tex]
Expanding this, we get:
[tex]\[ (2x^2 + 1)(2x^2 + 1) = 4x^4 + 2x^2 + 2x^2 + 1^2 = 4x^4 + 4x^2 + 1 \][/tex]
The right-hand side is:
[tex]\[ 4(x - 2) = 4x - 8 \][/tex]
So the equation now is:
[tex]\[ 4x^4 + 4x^2 + 1 = 4x - 8 \][/tex]
2. Reorganize the equation:
Move all the terms to one side to create a polynomial equal to zero:
[tex]\[ 4x^4 + 4x^2 + 1 - 4x + 8 = 0 \][/tex]
Simplify it:
[tex]\[ 4x^4 + 4x^2 - 4x + 9 = 0 \][/tex]
3. Find the roots of the polynomial:
Solving a polynomial of degree 4 can be complex because it involves finding the roots of a quartic equation. This typically involves methods beyond the quadratic formula and might require numerical methods or sophisticated algebraic techniques.
Using advanced algebraic methods or numerical approaches to solve the polynomial equation \(4x^4 + 4x^2 - 4x + 9 = 0\), we find the roots to be:
[tex]\[ x_1 = -\frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} + \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} - \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} \][/tex]
[tex]\[ x_2 = \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} - \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} + \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} \][/tex]
[tex]\[ x_3 = -\frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} - \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} - \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} \][/tex]
[tex]\[ x_4 = \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} + \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} + \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} \][/tex]
These expressions involve complex numbers and radicals, indicating the intricate nature of solving such quartic equations exactly.
1. Expand both sides:
On the left-hand side, we have a squared binomial:
[tex]\[ \left(2x^2 + 1\right)^2 = \left(2x^2 + 1\right) \left(2x^2 + 1\right) \][/tex]
Expanding this, we get:
[tex]\[ (2x^2 + 1)(2x^2 + 1) = 4x^4 + 2x^2 + 2x^2 + 1^2 = 4x^4 + 4x^2 + 1 \][/tex]
The right-hand side is:
[tex]\[ 4(x - 2) = 4x - 8 \][/tex]
So the equation now is:
[tex]\[ 4x^4 + 4x^2 + 1 = 4x - 8 \][/tex]
2. Reorganize the equation:
Move all the terms to one side to create a polynomial equal to zero:
[tex]\[ 4x^4 + 4x^2 + 1 - 4x + 8 = 0 \][/tex]
Simplify it:
[tex]\[ 4x^4 + 4x^2 - 4x + 9 = 0 \][/tex]
3. Find the roots of the polynomial:
Solving a polynomial of degree 4 can be complex because it involves finding the roots of a quartic equation. This typically involves methods beyond the quadratic formula and might require numerical methods or sophisticated algebraic techniques.
Using advanced algebraic methods or numerical approaches to solve the polynomial equation \(4x^4 + 4x^2 - 4x + 9 = 0\), we find the roots to be:
[tex]\[ x_1 = -\frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} + \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} - \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} \][/tex]
[tex]\[ x_2 = \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} - \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} + \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} \][/tex]
[tex]\[ x_3 = -\frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} - \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} - \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} \][/tex]
[tex]\[ x_4 = \frac{\sqrt{-\frac{4}{3} - 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3} + \frac{2}{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}} - \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}}{2} + \frac{\sqrt{-\frac{2}{3} + \frac{14}{9(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}} + 2(-\frac{133}{432} + \frac{7\sqrt{159}i}{144})^{1/3}}}{2} \][/tex]
These expressions involve complex numbers and radicals, indicating the intricate nature of solving such quartic equations exactly.
