Solve left(2900-2500-50xright)left(8+4xright)=5000

$$\left( 2900-2500-50x \right) \left( 8+4x \right) = 5000$$

Solve for x

$$x=3$$

Steps Using the Quadratic Formula

Subtract 2500 from 2900 to get 400.

$$\left(400-50x\right)\left(8+4x\right)=5000$$

Use the distributive property to multiply 400-50x by 8+4x and combine like terms.

$$3200+1200x-200x^{2}=5000$$

Subtract 5000 from both sides.

$$3200+1200x-200x^{2}-5000=0$$

Subtract 5000 from 3200 to get -1800.

$$-1800+1200x-200x^{2}=0$$

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.

$$-200x^{2}+1200x-1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -200 for a, 1200 for b, and -1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-200\right)\left(-1800\right)}}{2\left(-200\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-200\right)\left(-1800\right)}}{2\left(-200\right)}$$

Multiply -4 times -200.

$$x=\frac{-1200±\sqrt{1440000+800\left(-1800\right)}}{2\left(-200\right)}$$

Multiply 800 times -1800.

$$x=\frac{-1200±\sqrt{1440000-1440000}}{2\left(-200\right)}$$

Add 1440000 to -1440000.

$$x=\frac{-1200±\sqrt{0}}{2\left(-200\right)}$$

Take the square root of 0.

$$x=-\frac{1200}{2\left(-200\right)}$$

Multiply 2 times -200.

$$x=-\frac{1200}{-400}$$

Divide -1200 by -400.

$$x=3$$

Steps for Completing the Square

Subtract 2500 from 2900 to get 400.

$$\left(400-50x\right)\left(8+4x\right)=5000$$

Use the distributive property to multiply 400-50x by 8+4x and combine like terms.

$$3200+1200x-200x^{2}=5000$$

Subtract 3200 from both sides.

$$1200x-200x^{2}=5000-3200$$

Subtract 3200 from 5000 to get 1800.

$$1200x-200x^{2}=1800$$

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.

$$-200x^{2}+1200x=1800$$

Divide both sides by -200.

$$\frac{-200x^{2}+1200x}{-200}=\frac{1800}{-200}$$

Dividing by -200 undoes the multiplication by -200.

$$x^{2}+\frac{1200}{-200}x=\frac{1800}{-200}$$

Divide 1200 by -200.

$$x^{2}-6x=\frac{1800}{-200}$$

Divide 1800 by -200.

$$x^{2}-6x=-9$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-9+9$$

Add -9 to 9.

$$x^{2}-6x+9=0$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-3=0 x-3=0$$

Add 3 to both sides of the equation.

$$x=3 x=3$$

The equation is now solved. Solutions are the same.

$$x=3$$