Solve 1664=4+6(x-1)/2x

$$1664 = \frac { 4 + 6 ( x - 1 ) } { 2 } x$$

Solve for x

$$x = \frac{\sqrt{19969} + 1}{6} \approx 23.718618697$$

$$x=\frac{1-\sqrt{19969}}{6}\approx -23.385285364$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$3328=\left(4+6\left(x-1\right)\right)x$$

Use the distributive property to multiply 6 by x-1.

$$3328=\left(4+6x-6\right)x$$

Subtract 6 from 4 to get -2.

$$3328=\left(-2+6x\right)x$$

Use the distributive property to multiply -2+6x by x.

$$3328=-2x+6x^{2}$$

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

$$-2x+6x^{2}=3328$$

Subtract 3328 from both sides.

$$-2x+6x^{2}-3328=0$$

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.

$$6x^{2}-2x-3328=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -2 for b, and -3328 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 6\left(-3328\right)}}{2\times 6}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 6\left(-3328\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-2\right)±\sqrt{4-24\left(-3328\right)}}{2\times 6}$$

Multiply -24 times -3328.

$$x=\frac{-\left(-2\right)±\sqrt{4+79872}}{2\times 6}$$

Add 4 to 79872.

$$x=\frac{-\left(-2\right)±\sqrt{79876}}{2\times 6}$$

Take the square root of 79876.

$$x=\frac{-\left(-2\right)±2\sqrt{19969}}{2\times 6}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{19969}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{2±2\sqrt{19969}}{12}$$

Now solve the equation x=\frac{2±2\sqrt{19969}}{12} when ± is plus. Add 2 to 2\sqrt{19969}.

$$x=\frac{2\sqrt{19969}+2}{12}$$

Divide 2+2\sqrt{19969} by 12.

$$x=\frac{\sqrt{19969}+1}{6}$$

Now solve the equation x=\frac{2±2\sqrt{19969}}{12} when ± is minus. Subtract 2\sqrt{19969} from 2.

$$x=\frac{2-2\sqrt{19969}}{12}$$

Divide 2-2\sqrt{19969} by 12.

$$x=\frac{1-\sqrt{19969}}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{19969}+1}{6} x=\frac{1-\sqrt{19969}}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$3328=\left(4+6\left(x-1\right)\right)x$$

Use the distributive property to multiply 6 by x-1.

$$3328=\left(4+6x-6\right)x$$

Subtract 6 from 4 to get -2.

$$3328=\left(-2+6x\right)x$$

Use the distributive property to multiply -2+6x by x.

$$3328=-2x+6x^{2}$$

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

$$-2x+6x^{2}=3328$$

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.

$$6x^{2}-2x=3328$$

Divide both sides by 6.

$$\frac{6x^{2}-2x}{6}=\frac{3328}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{2}{6}\right)x=\frac{3328}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{3}x=\frac{3328}{6}$$

Reduce the fraction \frac{3328}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{3}x=\frac{1664}{3}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=\frac{1664}{3}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{1664}{3}+\frac{1}{36}$$

Add \frac{1664}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{19969}{36}$$

Factor x^{2}-\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{19969}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{19969}{36}}$$

Simplify.

$$x-\frac{1}{6}=\frac{\sqrt{19969}}{6} x-\frac{1}{6}=-\frac{\sqrt{19969}}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$x=\frac{\sqrt{19969}+1}{6} x=\frac{1-\sqrt{19969}}{6}$$