Solve y=2(5x^2+1)^3

$$y = 2 ( 5 x ^ { 2 } + 1 ) ^ { 3 }$$

Solve for y

$$y=2\left(5x^{2}+1\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x^{2}+1\right)^{3}.

$$y=2\left(125\left(x^{2}\right)^{3}+75\left(x^{2}\right)^{2}+15x^{2}+1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$y=2\left(125x^{6}+75\left(x^{2}\right)^{2}+15x^{2}+1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$y=2\left(125x^{6}+75x^{4}+15x^{2}+1\right)$$

Use the distributive property to multiply 2 by 125x^{6}+75x^{4}+15x^{2}+1.

$$y=250x^{6}+150x^{4}+30x^{2}+2$$

Solve for x

$$x=\frac{\sqrt{10\left(2^{\frac{2}{3}}\sqrt[3]{y}-2\right)}}{10}$$

$$x=-\frac{\sqrt{10\left(2^{\frac{2}{3}}\sqrt[3]{y}-2\right)}}{10}\text{, }y\geq 2$$