Solve (x-3)(x-5)=3

$$( x - 3 ) ( x - 5 ) = 3$$

Solve for x

$$x=2$$

$$x=6$$

Steps Using the Quadratic Formula

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

$$x^{2}-8x+15=3$$

Subtract 3 from both sides.

$$x^{2}-8x+15-3=0$$

Subtract 3 from 15 to get 12.

$$x^{2}-8x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2}$$

Add 64 to -48.

$$x=\frac{-\left(-8\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-8\right)±4}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±4}{2}$$

Now solve the equation x=\frac{8±4}{2} when ± is plus. Add 8 to 4.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{8±4}{2} when ± is minus. Subtract 4 from 8.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=6 x=2$$

Steps for Completing the Square

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

$$x^{2}-8x+15=3$$

Subtract 15 from both sides.

$$x^{2}-8x=3-15$$

Subtract 15 from 3 to get -12.

$$x^{2}-8x=-12$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-12+16$$

Add -12 to 16.

$$x^{2}-8x+16=4$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=2 x-4=-2$$

Add 4 to both sides of the equation.

$$x=6 x=2$$