Solve 1000=x/2(2(8)+(x-1)7)

$$1000= \frac{ x }{ 2 } (2(8)+(x-1)7)$$

Solve for x

$$x = \frac{\sqrt{56081} - 9}{14} \approx 16.272448087$$

$$x=\frac{-\sqrt{56081}-9}{14}\approx -17.558162373$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2000=x\left(2\times 8+\left(x-1\right)\times 7\right)$$

Multiply 2 and 8 to get 16.

$$2000=x\left(16+\left(x-1\right)\times 7\right)$$

Use the distributive property to multiply x-1 by 7.

$$2000=x\left(16+7x-7\right)$$

Subtract 7 from 16 to get 9.

$$2000=x\left(9+7x\right)$$

Use the distributive property to multiply x by 9+7x.

$$2000=9x+7x^{2}$$

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

$$9x+7x^{2}=2000$$

Subtract 2000 from both sides.

$$9x+7x^{2}-2000=0$$

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.

$$7x^{2}+9x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 9 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 7\left(-2000\right)}}{2\times 7}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 7\left(-2000\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-9±\sqrt{81-28\left(-2000\right)}}{2\times 7}$$

Multiply -28 times -2000.

$$x=\frac{-9±\sqrt{81+56000}}{2\times 7}$$

Add 81 to 56000.

$$x=\frac{-9±\sqrt{56081}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-9±\sqrt{56081}}{14}$$

Now solve the equation x=\frac{-9±\sqrt{56081}}{14} when ± is plus. Add -9 to \sqrt{56081}.

$$x=\frac{\sqrt{56081}-9}{14}$$

Now solve the equation x=\frac{-9±\sqrt{56081}}{14} when ± is minus. Subtract \sqrt{56081} from -9.

$$x=\frac{-\sqrt{56081}-9}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{56081}-9}{14} x=\frac{-\sqrt{56081}-9}{14}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2000=x\left(2\times 8+\left(x-1\right)\times 7\right)$$

Multiply 2 and 8 to get 16.

$$2000=x\left(16+\left(x-1\right)\times 7\right)$$

Use the distributive property to multiply x-1 by 7.

$$2000=x\left(16+7x-7\right)$$

Subtract 7 from 16 to get 9.

$$2000=x\left(9+7x\right)$$

Use the distributive property to multiply x by 9+7x.

$$2000=9x+7x^{2}$$

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

$$9x+7x^{2}=2000$$

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.

$$7x^{2}+9x=2000$$

Divide both sides by 7.

$$\frac{7x^{2}+9x}{7}=\frac{2000}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{9}{7}x=\frac{2000}{7}$$

Divide \frac{9}{7}, the coefficient of the x term, by 2 to get \frac{9}{14}. Then add the square of \frac{9}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{7}x+\left(\frac{9}{14}\right)^{2}=\frac{2000}{7}+\left(\frac{9}{14}\right)^{2}$$

Square \frac{9}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{7}x+\frac{81}{196}=\frac{2000}{7}+\frac{81}{196}$$

Add \frac{2000}{7} to \frac{81}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{7}x+\frac{81}{196}=\frac{56081}{196}$$

Factor x^{2}+\frac{9}{7}x+\frac{81}{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}.

$$\left(x+\frac{9}{14}\right)^{2}=\frac{56081}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{14}\right)^{2}}=\sqrt{\frac{56081}{196}}$$

Simplify.

$$x+\frac{9}{14}=\frac{\sqrt{56081}}{14} x+\frac{9}{14}=-\frac{\sqrt{56081}}{14}$$

Subtract \frac{9}{14} from both sides of the equation.

$$x=\frac{\sqrt{56081}-9}{14} x=\frac{-\sqrt{56081}-9}{14}$$