Solve 81x=200+7x^2

$$81x=200+7 { x }^{ 2 }$$

Solve for x

$$x=8$$

$$x = \frac{25}{7} = 3\frac{4}{7} \approx 3.571428571$$

Steps Using the Quadratic Formula

Subtract 200 from both sides.

$$81x-200=7x^{2}$$

Subtract 7x^{2} from both sides.

$$81x-200-7x^{2}=0$$

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.

$$-7x^{2}+81x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 81 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-81±\sqrt{81^{2}-4\left(-7\right)\left(-200\right)}}{2\left(-7\right)}$$

Square 81.

$$x=\frac{-81±\sqrt{6561-4\left(-7\right)\left(-200\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-81±\sqrt{6561+28\left(-200\right)}}{2\left(-7\right)}$$

Multiply 28 times -200.

$$x=\frac{-81±\sqrt{6561-5600}}{2\left(-7\right)}$$

Add 6561 to -5600.

$$x=\frac{-81±\sqrt{961}}{2\left(-7\right)}$$

Take the square root of 961.

$$x=\frac{-81±31}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-81±31}{-14}$$

Now solve the equation x=\frac{-81±31}{-14} when ± is plus. Add -81 to 31.

$$x=-\frac{50}{-14}$$

Reduce the fraction \frac{-50}{-14} to lowest terms by extracting and canceling out 2.

$$x=\frac{25}{7}$$

Now solve the equation x=\frac{-81±31}{-14} when ± is minus. Subtract 31 from -81.

$$x=-\frac{112}{-14}$$

Divide -112 by -14.

$$x=8$$

The equation is now solved.

$$x=\frac{25}{7} x=8$$

Steps for Completing the Square

Subtract 7x^{2} from both sides.

$$81x-7x^{2}=200$$

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.

$$-7x^{2}+81x=200$$

Divide both sides by -7.

$$\frac{-7x^{2}+81x}{-7}=\frac{200}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{81}{-7}x=\frac{200}{-7}$$

Divide 81 by -7.

$$x^{2}-\frac{81}{7}x=\frac{200}{-7}$$

Divide 200 by -7.

$$x^{2}-\frac{81}{7}x=-\frac{200}{7}$$

Divide -\frac{81}{7}, the coefficient of the x term, by 2 to get -\frac{81}{14}. Then add the square of -\frac{81}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{81}{7}x+\left(-\frac{81}{14}\right)^{2}=-\frac{200}{7}+\left(-\frac{81}{14}\right)^{2}$$

Square -\frac{81}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{81}{7}x+\frac{6561}{196}=-\frac{200}{7}+\frac{6561}{196}$$

Add -\frac{200}{7} to \frac{6561}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{81}{7}x+\frac{6561}{196}=\frac{961}{196}$$

Factor x^{2}-\frac{81}{7}x+\frac{6561}{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}.

$$\left(x-\frac{81}{14}\right)^{2}=\frac{961}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{81}{14}\right)^{2}}=\sqrt{\frac{961}{196}}$$

Simplify.

$$x-\frac{81}{14}=\frac{31}{14} x-\frac{81}{14}=-\frac{31}{14}$$

Add \frac{81}{14} to both sides of the equation.

$$x=8 x=\frac{25}{7}$$