Solve (x-3)(x-5)+2=0

$$( x - 3 ) ( x - 5 ) + 2 = 0$$

Solve for x (complex solution)

$$x=4+i$$

$$x=4-i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-3 by x-5 and combine like terms.

$$x^{2}-8x+15+2=0$$

Add 15 and 2 to get 17.

$$x^{2}-8x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 17}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 17}}{2}$$

Multiply -4 times 17.

$$x=\frac{-\left(-8\right)±\sqrt{64-68}}{2}$$

Add 64 to -68.

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

Take the square root of -4.

$$x=\frac{-\left(-8\right)±2i}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±2i}{2}$$

Now solve the equation x=\frac{8±2i}{2} when ± is plus. Add 8 to 2i.

$$x=\frac{8+2i}{2}$$

Divide 8+2i by 2.

$$x=4+i$$

Now solve the equation x=\frac{8±2i}{2} when ± is minus. Subtract 2i from 8.

$$x=\frac{8-2i}{2}$$

Divide 8-2i by 2.

$$x=4-i$$

The equation is now solved.

$$x=4+i x=4-i$$

Steps for Completing the Square

Use the distributive property to multiply x-3 by x-5 and combine like terms.

$$x^{2}-8x+15+2=0$$

Add 15 and 2 to get 17.

$$x^{2}-8x+17=0$$

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

$$x^{2}-8x=-17$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-17+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-17+16$$

Add -17 to 16.

$$x^{2}-8x+16=-1$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x-4=i x-4=-i$$

Add 4 to both sides of the equation.

$$x=4+i x=4-i$$