Solve 4*50=(40-x-25)(400+5x)

$$4 \times 50 = ( 40 - x - 25 ) ( 400 + 5 x )$$

Solve for x

$$x = \frac{3 \sqrt{985} - 65}{2} \approx 14.577064479$$

$$x=\frac{-3\sqrt{985}-65}{2}\approx -79.577064479$$

Steps Using the Quadratic Formula

Multiply 4 and 50 to get 200.

$$200=\left(40-x-25\right)\left(400+5x\right)$$

Subtract 25 from 40 to get 15.

$$200=\left(15-x\right)\left(400+5x\right)$$

Use the distributive property to multiply 15-x by 400+5x and combine like terms.

$$200=6000-325x-5x^{2}$$

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

$$6000-325x-5x^{2}=200$$

Subtract 200 from both sides.

$$6000-325x-5x^{2}-200=0$$

Subtract 200 from 6000 to get 5800.

$$5800-325x-5x^{2}=0$$

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.

$$-5x^{2}-325x+5800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -325 for b, and 5800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-325\right)±\sqrt{\left(-325\right)^{2}-4\left(-5\right)\times 5800}}{2\left(-5\right)}$$

Square -325.

$$x=\frac{-\left(-325\right)±\sqrt{105625-4\left(-5\right)\times 5800}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-325\right)±\sqrt{105625+20\times 5800}}{2\left(-5\right)}$$

Multiply 20 times 5800.

$$x=\frac{-\left(-325\right)±\sqrt{105625+116000}}{2\left(-5\right)}$$

Add 105625 to 116000.

$$x=\frac{-\left(-325\right)±\sqrt{221625}}{2\left(-5\right)}$$

Take the square root of 221625.

$$x=\frac{-\left(-325\right)±15\sqrt{985}}{2\left(-5\right)}$$

The opposite of -325 is 325.

$$x=\frac{325±15\sqrt{985}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{325±15\sqrt{985}}{-10}$$

Now solve the equation x=\frac{325±15\sqrt{985}}{-10} when ± is plus. Add 325 to 15\sqrt{985}.

$$x=\frac{15\sqrt{985}+325}{-10}$$

Divide 325+15\sqrt{985} by -10.

$$x=\frac{-3\sqrt{985}-65}{2}$$

Now solve the equation x=\frac{325±15\sqrt{985}}{-10} when ± is minus. Subtract 15\sqrt{985} from 325.

$$x=\frac{325-15\sqrt{985}}{-10}$$

Divide 325-15\sqrt{985} by -10.

$$x=\frac{3\sqrt{985}-65}{2}$$

The equation is now solved.

$$x=\frac{-3\sqrt{985}-65}{2} x=\frac{3\sqrt{985}-65}{2}$$

Steps for Completing the Square

Multiply 4 and 50 to get 200.

$$200=\left(40-x-25\right)\left(400+5x\right)$$

Subtract 25 from 40 to get 15.

$$200=\left(15-x\right)\left(400+5x\right)$$

Use the distributive property to multiply 15-x by 400+5x and combine like terms.

$$200=6000-325x-5x^{2}$$

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

$$6000-325x-5x^{2}=200$$

Subtract 6000 from both sides.

$$-325x-5x^{2}=200-6000$$

Subtract 6000 from 200 to get -5800.

$$-325x-5x^{2}=-5800$$

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.

$$-5x^{2}-325x=-5800$$

Divide both sides by -5.

$$\frac{-5x^{2}-325x}{-5}=-\frac{5800}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{325}{-5}\right)x=-\frac{5800}{-5}$$

Divide -325 by -5.

$$x^{2}+65x=-\frac{5800}{-5}$$

Divide -5800 by -5.

$$x^{2}+65x=1160$$

Divide 65, the coefficient of the x term, by 2 to get \frac{65}{2}. Then add the square of \frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+65x+\left(\frac{65}{2}\right)^{2}=1160+\left(\frac{65}{2}\right)^{2}$$

Square \frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+65x+\frac{4225}{4}=1160+\frac{4225}{4}$$

Add 1160 to \frac{4225}{4}.

$$x^{2}+65x+\frac{4225}{4}=\frac{8865}{4}$$

Factor x^{2}+65x+\frac{4225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{65}{2}\right)^{2}=\frac{8865}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{65}{2}\right)^{2}}=\sqrt{\frac{8865}{4}}$$

Simplify.

$$x+\frac{65}{2}=\frac{3\sqrt{985}}{2} x+\frac{65}{2}=-\frac{3\sqrt{985}}{2}$$

Subtract \frac{65}{2} from both sides of the equation.

$$x=\frac{3\sqrt{985}-65}{2} x=\frac{-3\sqrt{985}-65}{2}$$