Solve x^2-10x+25=3

$$x ^ { 2 } - 10 x + 25 = 3$$

Solve for x

$$x=\sqrt{3}+5\approx 6.732050808$$

$$x=5-\sqrt{3}\approx 3.267949192$$

Steps Using the Quadratic Formula

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.

$$x^{2}-10x+25=3$$

Subtract 3 from both sides of the equation.

$$x^{2}-10x+25-3=3-3$$

Subtracting 3 from itself leaves 0.

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

Subtract 3 from 25.

$$x^{2}-10x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 22}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 22}}{2}$$

Multiply -4 times 22.

$$x=\frac{-\left(-10\right)±\sqrt{100-88}}{2}$$

Add 100 to -88.

$$x=\frac{-\left(-10\right)±\sqrt{12}}{2}$$

Take the square root of 12.

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

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{3}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

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.

$$x^{2}-10x+25=3$$

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}.

$$\left(x-5\right)^{2}=3$$

Take the square root of both sides of the equation.

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

Simplify.

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

Add 5 to both sides of the equation.

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