Solve y=(-4/3x+4)^2+x^2

$$y = ( - \frac { 4 } { 3 } x + 4 ) ^ { 2 } + x ^ { 2 }$$

Solve for y

$$y=\frac{25x^{2}}{9}-\frac{32x}{3}+16$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{4}{3}x+4\right)^{2}.

$$y=\frac{16}{9}x^{2}-\frac{32}{3}x+16+x^{2}$$

Combine \frac{16}{9}x^{2} and x^{2} to get \frac{25}{9}x^{2}.

$$y=\frac{25}{9}x^{2}-\frac{32}{3}x+16$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{25y-144}+48}{25}$$

$$x=\frac{3\sqrt{25y-144}+48}{25}$$

Solve for x

$$x=\frac{-3\sqrt{25y-144}+48}{25}$$

$$x=\frac{3\sqrt{25y-144}+48}{25}\text{, }y\geq \frac{144}{25}$$