Solve [x-(5+3sqrt{3})][x-(5-3sqrt{3})]=0

$$[ x - ( 5 + 3 \sqrt { 3 } ) ] [ x - ( 5 - 3 \sqrt { 3 } ) ] = 0$$

Solve for x

$$x=5-3\sqrt{3}\approx -0.196152423$$

$$x=3\sqrt{3}+5\approx 10.196152423$$

Steps Using the Quadratic Formula

To find the opposite of 5+3\sqrt{3}, find the opposite of each term.

$$\left(x-5-3\sqrt{3}\right)\left(x-\left(5-3\sqrt{3}\right)\right)=0$$

To find the opposite of 5-3\sqrt{3}, find the opposite of each term.

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

Use the distributive property to multiply x-5-3\sqrt{3} by x-5+3\sqrt{3} and combine like terms.

$$x^{2}-10x+25-9\left(\sqrt{3}\right)^{2}=0$$

The square of \sqrt{3} is 3.

$$x^{2}-10x+25-9\times 3=0$$

Multiply -9 and 3 to get -27.

$$x^{2}-10x+25-27=0$$

Subtract 27 from 25 to get -2.

$$x^{2}-10x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-2\right)}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-\left(-10\right)±\sqrt{100+8}}{2}$$

Add 100 to 8.

$$x=\frac{-\left(-10\right)±\sqrt{108}}{2}$$

Take the square root of 108.

$$x=\frac{-\left(-10\right)±6\sqrt{3}}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±6\sqrt{3}}{2}$$

Now solve the equation x=\frac{10±6\sqrt{3}}{2} when ± is plus. Add 10 to 6\sqrt{3}.

$$x=\frac{6\sqrt{3}+10}{2}$$

Divide 10+6\sqrt{3} by 2.

$$x=3\sqrt{3}+5$$

Now solve the equation x=\frac{10±6\sqrt{3}}{2} when ± is minus. Subtract 6\sqrt{3} from 10.

$$x=\frac{10-6\sqrt{3}}{2}$$

Divide 10-6\sqrt{3} by 2.

$$x=5-3\sqrt{3}$$

The equation is now solved.

$$x=3\sqrt{3}+5 x=5-3\sqrt{3}$$

Steps for Completing the Square

To find the opposite of 5+3\sqrt{3}, find the opposite of each term.

$$\left(x-5-3\sqrt{3}\right)\left(x-\left(5-3\sqrt{3}\right)\right)=0$$

To find the opposite of 5-3\sqrt{3}, find the opposite of each term.

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

Use the distributive property to multiply x-5-3\sqrt{3} by x-5+3\sqrt{3} and combine like terms.

$$x^{2}-10x+25-9\left(\sqrt{3}\right)^{2}=0$$

The square of \sqrt{3} is 3.

$$x^{2}-10x+25-9\times 3=0$$

Multiply -9 and 3 to get -27.

$$x^{2}-10x+25-27=0$$

Subtract 27 from 25 to get -2.

$$x^{2}-10x-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$x^{2}-10x=2$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=2+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=2+25$$

Add 2 to 25.

$$x^{2}-10x+25=27$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=27$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{27}$$

Simplify.

$$x-5=3\sqrt{3} x-5=-3\sqrt{3}$$

Add 5 to both sides of the equation.

$$x=3\sqrt{3}+5 x=5-3\sqrt{3}$$