Solve 25x+35x(200-x)=6280

$$25x+35x(200-x)=6280$$

Solve for x

$$x = \frac{\sqrt{1938857} + 1405}{14} \approx 199.816318155$$

$$x=\frac{1405-\sqrt{1938857}}{14}\approx 0.897967559$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 35x by 200-x.

$$25x+7000x-35x^{2}=6280$$

Combine 25x and 7000x to get 7025x.

$$7025x-35x^{2}=6280$$

Subtract 6280 from both sides.

$$7025x-35x^{2}-6280=0$$

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.

$$-35x^{2}+7025x-6280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -35 for a, 7025 for b, and -6280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7025±\sqrt{7025^{2}-4\left(-35\right)\left(-6280\right)}}{2\left(-35\right)}$$

Square 7025.

$$x=\frac{-7025±\sqrt{49350625-4\left(-35\right)\left(-6280\right)}}{2\left(-35\right)}$$

Multiply -4 times -35.

$$x=\frac{-7025±\sqrt{49350625+140\left(-6280\right)}}{2\left(-35\right)}$$

Multiply 140 times -6280.

$$x=\frac{-7025±\sqrt{49350625-879200}}{2\left(-35\right)}$$

Add 49350625 to -879200.

$$x=\frac{-7025±\sqrt{48471425}}{2\left(-35\right)}$$

Take the square root of 48471425.

$$x=\frac{-7025±5\sqrt{1938857}}{2\left(-35\right)}$$

Multiply 2 times -35.

$$x=\frac{-7025±5\sqrt{1938857}}{-70}$$

Now solve the equation x=\frac{-7025±5\sqrt{1938857}}{-70} when ± is plus. Add -7025 to 5\sqrt{1938857}.

$$x=\frac{5\sqrt{1938857}-7025}{-70}$$

Divide -7025+5\sqrt{1938857} by -70.

$$x=\frac{1405-\sqrt{1938857}}{14}$$

Now solve the equation x=\frac{-7025±5\sqrt{1938857}}{-70} when ± is minus. Subtract 5\sqrt{1938857} from -7025.

$$x=\frac{-5\sqrt{1938857}-7025}{-70}$$

Divide -7025-5\sqrt{1938857} by -70.

$$x=\frac{\sqrt{1938857}+1405}{14}$$

The equation is now solved.

$$x=\frac{1405-\sqrt{1938857}}{14} x=\frac{\sqrt{1938857}+1405}{14}$$

Steps for Completing the Square

Use the distributive property to multiply 35x by 200-x.

$$25x+7000x-35x^{2}=6280$$

Combine 25x and 7000x to get 7025x.

$$7025x-35x^{2}=6280$$

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.

$$-35x^{2}+7025x=6280$$

Divide both sides by -35.

$$\frac{-35x^{2}+7025x}{-35}=\frac{6280}{-35}$$

Dividing by -35 undoes the multiplication by -35.

$$x^{2}+\frac{7025}{-35}x=\frac{6280}{-35}$$

Reduce the fraction \frac{7025}{-35} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{1405}{7}x=\frac{6280}{-35}$$

Reduce the fraction \frac{6280}{-35} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{1405}{7}x=-\frac{1256}{7}$$

Divide -\frac{1405}{7}, the coefficient of the x term, by 2 to get -\frac{1405}{14}. Then add the square of -\frac{1405}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1405}{7}x+\left(-\frac{1405}{14}\right)^{2}=-\frac{1256}{7}+\left(-\frac{1405}{14}\right)^{2}$$

Square -\frac{1405}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1405}{7}x+\frac{1974025}{196}=-\frac{1256}{7}+\frac{1974025}{196}$$

Add -\frac{1256}{7} to \frac{1974025}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1405}{7}x+\frac{1974025}{196}=\frac{1938857}{196}$$

Factor x^{2}-\frac{1405}{7}x+\frac{1974025}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1405}{14}\right)^{2}=\frac{1938857}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1405}{14}\right)^{2}}=\sqrt{\frac{1938857}{196}}$$

Simplify.

$$x-\frac{1405}{14}=\frac{\sqrt{1938857}}{14} x-\frac{1405}{14}=-\frac{\sqrt{1938857}}{14}$$

Add \frac{1405}{14} to both sides of the equation.

$$x=\frac{\sqrt{1938857}+1405}{14} x=\frac{1405-\sqrt{1938857}}{14}$$