Question

Use U-Substitution to solve the following polynomial. 4x^4 + 2x^2 - 12 = 0

Answer 1

ANSWER

$x=\sqrt[]{\frac{3}{2}},x=-\sqrt[]{\frac{3}{2}},x=\sqrt[]{2}i,x=-\sqrt[]{2}i$

EXPLANATION

We want to solve the given polynomial by U-Substitution:

$4x^4+2x^2-12=0$

To do this, we make the following substitution:

$u=x^2$

The polynomial then becomes:

$4u^2+2u-12=0$

Solve the quadratic equation by factorization:

$\begin{gathered} 4u^2+8u-6u-12=0 \\ \Rightarrow4u(u+2)-6(u+2)=0 \\ \Rightarrow(4u-6)(u+2)=0 \\ \Rightarrow4u-6=0;u+2=0 \\ \Rightarrow4u=6;u=-2 \\ u=\frac{6}{4}=\frac{3}{2};u=-2 \end{gathered}$

Recall that:

$\begin{gathered} u=x^2 \\ \Rightarrow x=\pm\sqrt[]{u} \end{gathered}$

Therefore, we have that:

$\begin{gathered} x=\pm\sqrt[]{\frac{3}{2}};x=\pm\sqrt[]{-2}=\pm\sqrt[]{2\cdot-1} \\ x=\pm\sqrt[]{\frac{3}{2}};x=\pm\sqrt[]{2}i \\ \Rightarrow x=\sqrt[]{\frac{3}{2}},x=-\sqrt[]{\frac{3}{2}},x=\sqrt[]{2}i,x=-\sqrt[]{2}i \end{gathered}$

Those are the solutions of the polynomial.