Solve 10x-60+6sqrt{4+(6-x^2)}=0

$$10 x - 60 + 6 \sqrt { 4 + ( 6 - x ^ { 2 } ) } = 0$$

Solve for x (complex solution)

$$x=\frac{-6\sqrt{35}i+75}{17}\approx 4.411764706-2.088028159i$$

Solution Steps

Subtract 10x-60 from both sides of the equation.

$$6\sqrt{4+6-x^{2}}=-\left(10x-60\right)$$

Add 4 and 6 to get 10.

$$6\sqrt{10-x^{2}}=-\left(10x-60\right)$$

To find the opposite of 10x-60, find the opposite of each term.

$$6\sqrt{10-x^{2}}=-10x+60$$

Square both sides of the equation.

$$\left(6\sqrt{10-x^{2}}\right)^{2}=\left(-10x+60\right)^{2}$$

Expand \left(6\sqrt{10-x^{2}}\right)^{2}.

$$6^{2}\left(\sqrt{10-x^{2}}\right)^{2}=\left(-10x+60\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$36\left(\sqrt{10-x^{2}}\right)^{2}=\left(-10x+60\right)^{2}$$

Calculate \sqrt{10-x^{2}} to the power of 2 and get 10-x^{2}.

$$36\left(10-x^{2}\right)=\left(-10x+60\right)^{2}$$

Use the distributive property to multiply 36 by 10-x^{2}.

$$360-36x^{2}=\left(-10x+60\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-10x+60\right)^{2}.

$$360-36x^{2}=100x^{2}-1200x+3600$$

Subtract 100x^{2} from both sides.

$$360-36x^{2}-100x^{2}=-1200x+3600$$

Combine -36x^{2} and -100x^{2} to get -136x^{2}.

$$360-136x^{2}=-1200x+3600$$

Add 1200x to both sides.

$$360-136x^{2}+1200x=3600$$

Subtract 3600 from both sides.

$$360-136x^{2}+1200x-3600=0$$

Subtract 3600 from 360 to get -3240.

$$-3240-136x^{2}+1200x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-136x^{2}+1200x-3240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -136 for a, 1200 for b, and -3240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-136\right)\left(-3240\right)}}{2\left(-136\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-136\right)\left(-3240\right)}}{2\left(-136\right)}$$

Multiply -4 times -136.

$$x=\frac{-1200±\sqrt{1440000+544\left(-3240\right)}}{2\left(-136\right)}$$

Multiply 544 times -3240.

$$x=\frac{-1200±\sqrt{1440000-1762560}}{2\left(-136\right)}$$

Add 1440000 to -1762560.

$$x=\frac{-1200±\sqrt{-322560}}{2\left(-136\right)}$$

Take the square root of -322560.

$$x=\frac{-1200±96\sqrt{35}i}{2\left(-136\right)}$$

Multiply 2 times -136.

$$x=\frac{-1200±96\sqrt{35}i}{-272}$$

Now solve the equation x=\frac{-1200±96\sqrt{35}i}{-272} when ± is plus. Add -1200 to 96i\sqrt{35}.

$$x=\frac{-1200+96\sqrt{35}i}{-272}$$

Divide -1200+96i\sqrt{35} by -272.

$$x=\frac{-6\sqrt{35}i+75}{17}$$

Now solve the equation x=\frac{-1200±96\sqrt{35}i}{-272} when ± is minus. Subtract 96i\sqrt{35} from -1200.

$$x=\frac{-96\sqrt{35}i-1200}{-272}$$

Divide -1200-96i\sqrt{35} by -272.

$$x=\frac{75+6\sqrt{35}i}{17}$$

The equation is now solved.

$$x=\frac{-6\sqrt{35}i+75}{17} x=\frac{75+6\sqrt{35}i}{17}$$

Substitute \frac{-6\sqrt{35}i+75}{17} for x in the equation 10x-60+6\sqrt{4+6-x^{2}}=0.

$$10\times \frac{-6\sqrt{35}i+75}{17}-60+6\sqrt{4+6-\left(\frac{-6\sqrt{35}i+75}{17}\right)^{2}}=0$$

Simplify. The value x=\frac{-6\sqrt{35}i+75}{17} satisfies the equation.

$$0=0$$

Substitute \frac{75+6\sqrt{35}i}{17} for x in the equation 10x-60+6\sqrt{4+6-x^{2}}=0.

$$10\times \frac{75+6\sqrt{35}i}{17}-60+6\sqrt{4+6-\left(\frac{75+6\sqrt{35}i}{17}\right)^{2}}=0$$

Simplify. The value x=\frac{75+6\sqrt{35}i}{17} does not satisfy the equation.

$$-\frac{540}{17}+\frac{120}{17}i\times 35^{\frac{1}{2}}=0$$

Equation 6\sqrt{10-x^{2}}=60-10x has a unique solution.

$$x=\frac{-6\sqrt{35}i+75}{17}$$