Solve 80*20=(65+x)(25-x)

$$80 \times 20=(65+x)(25-x)$$

Solve for x

$$x=5\sqrt{17}-20\approx 0.615528128$$

$$x=-5\sqrt{17}-20\approx -40.615528128$$

Steps Using the Quadratic Formula

Multiply 80 and 20 to get 1600.

$$1600=\left(65+x\right)\left(25-x\right)$$

Use the distributive property to multiply 65+x by 25-x and combine like terms.

$$1600=1625-40x-x^{2}$$

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

$$1625-40x-x^{2}=1600$$

Subtract 1600 from both sides.

$$1625-40x-x^{2}-1600=0$$

Subtract 1600 from 1625 to get 25.

$$25-40x-x^{2}=0$$

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.

$$-x^{2}-40x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -40 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-40\right)±\sqrt{1600+4\times 25}}{2\left(-1\right)}$$

Multiply 4 times 25.

$$x=\frac{-\left(-40\right)±\sqrt{1600+100}}{2\left(-1\right)}$$

Add 1600 to 100.

$$x=\frac{-\left(-40\right)±\sqrt{1700}}{2\left(-1\right)}$$

Take the square root of 1700.

$$x=\frac{-\left(-40\right)±10\sqrt{17}}{2\left(-1\right)}$$

The opposite of -40 is 40.

$$x=\frac{40±10\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{40±10\sqrt{17}}{-2}$$

Now solve the equation x=\frac{40±10\sqrt{17}}{-2} when ± is plus. Add 40 to 10\sqrt{17}.

$$x=\frac{10\sqrt{17}+40}{-2}$$

Divide 40+10\sqrt{17} by -2.

$$x=-5\sqrt{17}-20$$

Now solve the equation x=\frac{40±10\sqrt{17}}{-2} when ± is minus. Subtract 10\sqrt{17} from 40.

$$x=\frac{40-10\sqrt{17}}{-2}$$

Divide 40-10\sqrt{17} by -2.

$$x=5\sqrt{17}-20$$

The equation is now solved.

$$x=-5\sqrt{17}-20 x=5\sqrt{17}-20$$

Steps for Completing the Square

Multiply 80 and 20 to get 1600.

$$1600=\left(65+x\right)\left(25-x\right)$$

Use the distributive property to multiply 65+x by 25-x and combine like terms.

$$1600=1625-40x-x^{2}$$

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

$$1625-40x-x^{2}=1600$$

Subtract 1625 from both sides.

$$-40x-x^{2}=1600-1625$$

Subtract 1625 from 1600 to get -25.

$$-40x-x^{2}=-25$$

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.

$$-x^{2}-40x=-25$$

Divide both sides by -1.

$$\frac{-x^{2}-40x}{-1}=-\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{40}{-1}\right)x=-\frac{25}{-1}$$

Divide -40 by -1.

$$x^{2}+40x=-\frac{25}{-1}$$

Divide -25 by -1.

$$x^{2}+40x=25$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+40x+20^{2}=25+20^{2}$$

Square 20.

$$x^{2}+40x+400=25+400$$

Add 25 to 400.

$$x^{2}+40x+400=425$$

Factor x^{2}+40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+20\right)^{2}=425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{425}$$

Simplify.

$$x+20=5\sqrt{17} x+20=-5\sqrt{17}$$

Subtract 20 from both sides of the equation.

$$x=5\sqrt{17}-20 x=-5\sqrt{17}-20$$