Solve the following equation for x

Question

Solve the following equation for x

$$10^{\log_{10} {\sqrt2}} - e^{\ln4} = {\sqrt2}{x^2} - 10^{\log_{10} x} -3$$

Answer 1

As You said in the comment..So the question is equivalent to $$\sqrt2-4=\sqrt2x^2-x-3$$

A $(a^{\log_bc})=c^{\log_ba}$

So your equation is $$2x^2-\sqrt2x+(\sqrt2-2)=0$$ Using quadratic formula gives the solution....

$$\frac{\sqrt2\pm\sqrt{18-2(\sqrt32)}}{4}=\frac{\sqrt2\pm(4-\sqrt2)}{4}$$

Answer 2

Hint

Use that $$a^{\log_a b}=b$$

and then your equation becomes:

$$\sqrt{2}-4=\sqrt{2}x^2-x-3\to \sqrt{2}x^2-x+1-\sqrt{2}=0$$

Can you finish?