Solve 225=(x-10)(40-x)

$$225 = ( x - 10 ) ( 40 - x )$$

Solve for x

$$x=25$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by 40-x and combine like terms.

$$225=50x-x^{2}-400$$

Swap sides so that all variable terms are on the left hand side.

$$50x-x^{2}-400=225$$

Subtract 225 from both sides.

$$50x-x^{2}-400-225=0$$

Subtract 225 from -400 to get -625.

$$50x-x^{2}-625=0$$

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.

$$-x^{2}+50x-625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 50 for b, and -625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-1\right)\left(-625\right)}}{2\left(-1\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-1\right)\left(-625\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-50±\sqrt{2500+4\left(-625\right)}}{2\left(-1\right)}$$

Multiply 4 times -625.

$$x=\frac{-50±\sqrt{2500-2500}}{2\left(-1\right)}$$

Add 2500 to -2500.

$$x=\frac{-50±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{50}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{50}{-2}$$

Divide -50 by -2.

$$x=25$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by 40-x and combine like terms.

$$225=50x-x^{2}-400$$

Swap sides so that all variable terms are on the left hand side.

$$50x-x^{2}-400=225$$

Add 400 to both sides.

$$50x-x^{2}=225+400$$

Add 225 and 400 to get 625.

$$50x-x^{2}=625$$

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.

$$-x^{2}+50x=625$$

Divide both sides by -1.

$$\frac{-x^{2}+50x}{-1}=\frac{625}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{50}{-1}x=\frac{625}{-1}$$

Divide 50 by -1.

$$x^{2}-50x=\frac{625}{-1}$$

Divide 625 by -1.

$$x^{2}-50x=-625$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-625+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-625+625$$

Add -625 to 625.

$$x^{2}-50x+625=0$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-25=0 x-25=0$$

Add 25 to both sides of the equation.

$$x=25 x=25$$

The equation is now solved. Solutions are the same.

$$x=25$$