Solve x^2-10x+20-3=0

$$x ^ { 2 } - 10 x + 20 - 3 = 0$$

Solve for x

$$x=2\sqrt{2}+5\approx 7.828427125$$

$$x=5-2\sqrt{2}\approx 2.171572875$$

Steps Using the Quadratic Formula

Subtract 3 from 20 to get 17.

$$x^{2}-10x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 17}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 17}}{2}$$

Multiply -4 times 17.

$$x=\frac{-\left(-10\right)±\sqrt{100-68}}{2}$$

Add 100 to -68.

$$x=\frac{-\left(-10\right)±\sqrt{32}}{2}$$

Take the square root of 32.

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

The opposite of -10 is 10.

$$x=\frac{10±4\sqrt{2}}{2}$$

Now solve the equation x=\frac{10±4\sqrt{2}}{2} when ± is plus. Add 10 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+10}{2}$$

Divide 10+4\sqrt{2} by 2.

$$x=2\sqrt{2}+5$$

Now solve the equation x=\frac{10±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from 10.

$$x=\frac{10-4\sqrt{2}}{2}$$

Divide 10-4\sqrt{2} by 2.

$$x=5-2\sqrt{2}$$

The equation is now solved.

$$x=2\sqrt{2}+5 x=5-2\sqrt{2}$$

Steps for Completing the Square

Subtract 3 from 20 to get 17.

$$x^{2}-10x+17=0$$

Subtract 17 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-10x=-17$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-17+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-17+25$$

Add -17 to 25.

$$x^{2}-10x+25=8$$

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}.

$$\left(x-5\right)^{2}=8$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-5=2\sqrt{2} x-5=-2\sqrt{2}$$

Add 5 to both sides of the equation.

$$x=2\sqrt{2}+5 x=5-2\sqrt{2}$$