Solve (x-4)(x-6)=225

$$( x - 4 ) ( x - 6 ) = 225$$

Solve for x

$$x=\sqrt{226}+5\approx 20.033296378$$

$$x=5-\sqrt{226}\approx -10.033296378$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-4 by x-6 and combine like terms.

$$x^{2}-10x+24=225$$

Subtract 225 from both sides.

$$x^{2}-10x+24-225=0$$

Subtract 225 from 24 to get -201.

$$x^{2}-10x-201=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -201 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-201\right)}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-201\right)}}{2}$$

Multiply -4 times -201.

$$x=\frac{-\left(-10\right)±\sqrt{100+804}}{2}$$

Add 100 to 804.

$$x=\frac{-\left(-10\right)±\sqrt{904}}{2}$$

Take the square root of 904.

$$x=\frac{-\left(-10\right)±2\sqrt{226}}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{226}}{2}$$

Now solve the equation x=\frac{10±2\sqrt{226}}{2} when ± is plus. Add 10 to 2\sqrt{226}.

$$x=\frac{2\sqrt{226}+10}{2}$$

Divide 10+2\sqrt{226} by 2.

$$x=\sqrt{226}+5$$

Now solve the equation x=\frac{10±2\sqrt{226}}{2} when ± is minus. Subtract 2\sqrt{226} from 10.

$$x=\frac{10-2\sqrt{226}}{2}$$

Divide 10-2\sqrt{226} by 2.

$$x=5-\sqrt{226}$$

The equation is now solved.

$$x=\sqrt{226}+5 x=5-\sqrt{226}$$

Steps for Completing the Square

Use the distributive property to multiply x-4 by x-6 and combine like terms.

$$x^{2}-10x+24=225$$

Subtract 24 from both sides.

$$x^{2}-10x=225-24$$

Subtract 24 from 225 to get 201.

$$x^{2}-10x=201$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=201+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=201+25$$

Add 201 to 25.

$$x^{2}-10x+25=226$$

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}.

$$\left(x-5\right)^{2}=226$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{226}$$

Simplify.

$$x-5=\sqrt{226} x-5=-\sqrt{226}$$

Add 5 to both sides of the equation.

$$x=\sqrt{226}+5 x=5-\sqrt{226}$$