Solve (5+sqrt{2})x^2-(4+sqrt{3})x+8+2sqrt{3}=0

$$( 5 + \sqrt { 2 } ) x ^ { 2 } - ( 4 + \sqrt { 3 } ) x + 8 + 2 \sqrt { 3 } = 0$$

Solve for x (complex solution)

$$x=\frac{\left(5-\sqrt{2}\right)\left(\sqrt{3}+4+i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}\right)}{46}\approx 0.446824131+1.260017746i$$

$$x=\frac{\left(5-\sqrt{2}\right)\left(-i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}+\sqrt{3}+4\right)}{46}\approx 0.446824131-1.260017746i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5+\sqrt{2} by x^{2}.

$$5x^{2}+\sqrt{2}x^{2}-\left(4+\sqrt{3}\right)x+8+2\sqrt{3}=0$$

Use the distributive property to multiply 4+\sqrt{3} by x.

$$5x^{2}+\sqrt{2}x^{2}-\left(4x+\sqrt{3}x\right)+8+2\sqrt{3}=0$$

To find the opposite of 4x+\sqrt{3}x, find the opposite of each term.

$$5x^{2}+\sqrt{2}x^{2}-4x-\sqrt{3}x+8+2\sqrt{3}=0$$

Combine all terms containing x.

$$\left(5+\sqrt{2}\right)x^{2}+\left(-4-\sqrt{3}\right)x+8+2\sqrt{3}=0$$

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.

$$\left(\sqrt{2}+5\right)x^{2}+\left(-\sqrt{3}-4\right)x+2\sqrt{3}+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5+\sqrt{2} for a, -4-\sqrt{3} for b, and 8+2\sqrt{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±\sqrt{\left(-\sqrt{3}-4\right)^{2}-4\left(\sqrt{2}+5\right)\left(2\sqrt{3}+8\right)}}{2\left(\sqrt{2}+5\right)}$$

Square -4-\sqrt{3}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±\sqrt{8\sqrt{3}+19-4\left(\sqrt{2}+5\right)\left(2\sqrt{3}+8\right)}}{2\left(\sqrt{2}+5\right)}$$

Multiply -4 times 5+\sqrt{2}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±\sqrt{8\sqrt{3}+19+\left(-4\sqrt{2}-20\right)\left(2\sqrt{3}+8\right)}}{2\left(\sqrt{2}+5\right)}$$

Multiply -20-4\sqrt{2} times 8+2\sqrt{3}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±\sqrt{8\sqrt{3}+19-8\sqrt{6}-32\sqrt{2}-40\sqrt{3}-160}}{2\left(\sqrt{2}+5\right)}$$

Add 19+8\sqrt{3} to -160-40\sqrt{3}-32\sqrt{2}-8\sqrt{6}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±\sqrt{-8\sqrt{6}-32\sqrt{2}-32\sqrt{3}-141}}{2\left(\sqrt{2}+5\right)}$$

Take the square root of -141-32\sqrt{3}-32\sqrt{2}-8\sqrt{6}.

$$x=\frac{-\left(-\sqrt{3}-4\right)±i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}}{2\left(\sqrt{2}+5\right)}$$

The opposite of -4-\sqrt{3} is 4+\sqrt{3}.

$$x=\frac{\sqrt{3}+4±i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}}{2\left(\sqrt{2}+5\right)}$$

Multiply 2 times 5+\sqrt{2}.

$$x=\frac{\sqrt{3}+4±i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}}{2\sqrt{2}+10}$$

Now solve the equation x=\frac{\sqrt{3}+4±i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}}{2\sqrt{2}+10} when ± is plus. Add 4+\sqrt{3} to i\sqrt{141+32\sqrt{3}+32\sqrt{2}+8\sqrt{6}}.

$$x=\frac{\sqrt{3}+4+i\sqrt{8\sqrt{6}+32\sqrt{3}+2^{\frac{11}{2}}+141}}{2\sqrt{2}+10}$$

Divide 4+\sqrt{3}+i\sqrt{141+32\sqrt{3}+2^{\frac{11}{2}}+8\sqrt{6}} by 10+2\sqrt{2}.

$$x=-\frac{\left(\sqrt{2}-5\right)\left(\sqrt{3}+4+i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}\right)}{46}$$

Now solve the equation x=\frac{\sqrt{3}+4±i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}}{2\sqrt{2}+10} when ± is minus. Subtract i\sqrt{141+32\sqrt{3}+32\sqrt{2}+8\sqrt{6}} from 4+\sqrt{3}.

$$x=\frac{-i\sqrt{8\sqrt{6}+32\sqrt{3}+2^{\frac{11}{2}}+141}+\sqrt{3}+4}{2\sqrt{2}+10}$$

Divide 4+\sqrt{3}-i\sqrt{141+32\sqrt{3}+2^{\frac{11}{2}}+8\sqrt{6}} by 10+2\sqrt{2}.

$$x=-\frac{\left(\sqrt{2}-5\right)\left(-i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}+\sqrt{3}+4\right)}{46}$$

The equation is now solved.

$$x=-\frac{\left(\sqrt{2}-5\right)\left(\sqrt{3}+4+i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}\right)}{46} x=-\frac{\left(\sqrt{2}-5\right)\left(-i\sqrt{8\sqrt{6}+32\sqrt{2}+32\sqrt{3}+141}+\sqrt{3}+4\right)}{46}$$

Steps for Completing the Square

Use the distributive property to multiply 5+\sqrt{2} by x^{2}.

$$5x^{2}+\sqrt{2}x^{2}-\left(4+\sqrt{3}\right)x+8+2\sqrt{3}=0$$

Use the distributive property to multiply 4+\sqrt{3} by x.

$$5x^{2}+\sqrt{2}x^{2}-\left(4x+\sqrt{3}x\right)+8+2\sqrt{3}=0$$

To find the opposite of 4x+\sqrt{3}x, find the opposite of each term.

$$5x^{2}+\sqrt{2}x^{2}-4x-\sqrt{3}x+8+2\sqrt{3}=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$5x^{2}+\sqrt{2}x^{2}-4x-\sqrt{3}x+2\sqrt{3}=-8$$

Subtract 2\sqrt{3} from both sides.

$$5x^{2}+\sqrt{2}x^{2}-4x-\sqrt{3}x=-8-2\sqrt{3}$$

Combine all terms containing x.

$$\left(5+\sqrt{2}\right)x^{2}+\left(-4-\sqrt{3}\right)x=-8-2\sqrt{3}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\left(\sqrt{2}+5\right)x^{2}+\left(-\sqrt{3}-4\right)x=-2\sqrt{3}-8$$

Divide both sides by 5+\sqrt{2}.

$$\frac{\left(\sqrt{2}+5\right)x^{2}+\left(-\sqrt{3}-4\right)x}{\sqrt{2}+5}=\frac{-2\sqrt{3}-8}{\sqrt{2}+5}$$

Dividing by 5+\sqrt{2} undoes the multiplication by 5+\sqrt{2}.

$$x^{2}+\frac{-\sqrt{3}-4}{\sqrt{2}+5}x=\frac{-2\sqrt{3}-8}{\sqrt{2}+5}$$

Divide -4-\sqrt{3} by 5+\sqrt{2}.

$$x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x=\frac{-2\sqrt{3}-8}{\sqrt{2}+5}$$

Divide -8-2\sqrt{3} by 5+\sqrt{2}.

$$x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x=\frac{2\sqrt{6}+8\sqrt{2}-10\sqrt{3}-40}{23}$$

Divide \frac{4\sqrt{2}-20+\sqrt{6}-5\sqrt{3}}{23}, the coefficient of the x term, by 2 to get \frac{2\sqrt{2}}{23}-\frac{10}{23}+\frac{\sqrt{6}}{46}-\frac{5\sqrt{3}}{46}. Then add the square of \frac{2\sqrt{2}}{23}-\frac{10}{23}+\frac{\sqrt{6}}{46}-\frac{5\sqrt{3}}{46} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x+\left(\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}\right)^{2}=\frac{2\sqrt{6}+8\sqrt{2}-10\sqrt{3}-40}{23}+\left(\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}\right)^{2}$$

Square \frac{2\sqrt{2}}{23}-\frac{10}{23}+\frac{\sqrt{6}}{46}-\frac{5\sqrt{3}}{46}.

$$x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x+\frac{54\sqrt{3}}{529}-\frac{20\sqrt{6}}{529}-\frac{190\sqrt{2}}{2116}+\frac{513}{2116}=\frac{2\sqrt{6}+8\sqrt{2}-10\sqrt{3}-40}{23}+\frac{54\sqrt{3}}{529}-\frac{20\sqrt{6}}{529}-\frac{190\sqrt{2}}{2116}+\frac{513}{2116}$$

Add \frac{8\sqrt{2}-40+2\sqrt{6}-10\sqrt{3}}{23} to \frac{513}{2116}-\frac{20\sqrt{6}}{529}+\frac{54\sqrt{3}}{529}-\frac{190\sqrt{2}}{2116}.

$$x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x+\frac{54\sqrt{3}}{529}-\frac{20\sqrt{6}}{529}-\frac{190\sqrt{2}}{2116}+\frac{513}{2116}=\frac{2\times 2^{\frac{5}{2}}}{23}+\frac{26\sqrt{6}}{529}-\frac{95\sqrt{2}}{1058}-\frac{176\sqrt{3}}{529}-\frac{3167}{2116}$$

Factor x^{2}+\frac{\sqrt{6}+4\sqrt{2}-5\sqrt{3}-20}{23}x+\frac{54\sqrt{3}}{529}-\frac{20\sqrt{6}}{529}-\frac{190\sqrt{2}}{2116}+\frac{513}{2116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}\right)^{2}=\frac{2\times 2^{\frac{5}{2}}}{23}+\frac{26\sqrt{6}}{529}-\frac{95\sqrt{2}}{1058}-\frac{176\sqrt{3}}{529}-\frac{3167}{2116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}\right)^{2}}=\sqrt{\frac{2\times 2^{\frac{5}{2}}}{23}+\frac{26\sqrt{6}}{529}-\frac{95\sqrt{2}}{1058}-\frac{176\sqrt{3}}{529}-\frac{3167}{2116}}$$

Simplify.

$$x+\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}=\frac{i\sqrt{704\sqrt{3}+3167-104\sqrt{6}-546\sqrt{2}}}{46} x+\frac{\sqrt{6}}{46}+\frac{2\sqrt{2}}{23}-\frac{5\sqrt{3}}{46}-\frac{10}{23}=-\frac{i\sqrt{704\sqrt{3}+3167-104\sqrt{6}-546\sqrt{2}}}{46}$$

Subtract \frac{2\sqrt{2}}{23}-\frac{10}{23}+\frac{\sqrt{6}}{46}-\frac{5\sqrt{3}}{46} from both sides of the equation.

$$x=\frac{i\sqrt{704\sqrt{3}+3167-104\sqrt{6}-546\sqrt{2}}}{46}+\frac{5\sqrt{3}}{46}-\frac{\sqrt{6}}{46}-\frac{2^{\frac{5}{2}}}{46}+\frac{10}{23} x=-\frac{i\sqrt{704\sqrt{3}+3167-104\sqrt{6}-546\sqrt{2}}}{46}+\frac{5\sqrt{3}}{46}-\frac{\sqrt{6}}{46}-\frac{2^{\frac{5}{2}}}{46}+\frac{10}{23}$$