Solve 2x/1-3x=8x-16/16x-11

$$\frac { 2 x } { 1 - 3 x } = \frac { 8 x - 16 } { 16 x - 11 }$$

Solve for x

$$x=\frac{1}{4}=0.25$$

$$x = \frac{8}{7} = 1\frac{1}{7} \approx 1.142857143$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{3},\frac{11}{16} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(16x-11\right), the least common multiple of 1-3x,16x-11.

$$\left(11-16x\right)\times 2x=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 11-16x by 2.

$$\left(22-32x\right)x=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 22-32x by x.

$$22x-32x^{2}=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 3x-1 by 8x-16 and combine like terms.

$$22x-32x^{2}=24x^{2}-56x+16$$

Subtract 24x^{2} from both sides.

$$22x-32x^{2}-24x^{2}=-56x+16$$

Combine -32x^{2} and -24x^{2} to get -56x^{2}.

$$22x-56x^{2}=-56x+16$$

Add 56x to both sides.

$$22x-56x^{2}+56x=16$$

Combine 22x and 56x to get 78x.

$$78x-56x^{2}=16$$

Subtract 16 from both sides.

$$78x-56x^{2}-16=0$$

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.

$$-56x^{2}+78x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -56 for a, 78 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-78±\sqrt{78^{2}-4\left(-56\right)\left(-16\right)}}{2\left(-56\right)}$$

Square 78.

$$x=\frac{-78±\sqrt{6084-4\left(-56\right)\left(-16\right)}}{2\left(-56\right)}$$

Multiply -4 times -56.

$$x=\frac{-78±\sqrt{6084+224\left(-16\right)}}{2\left(-56\right)}$$

Multiply 224 times -16.

$$x=\frac{-78±\sqrt{6084-3584}}{2\left(-56\right)}$$

Add 6084 to -3584.

$$x=\frac{-78±\sqrt{2500}}{2\left(-56\right)}$$

Take the square root of 2500.

$$x=\frac{-78±50}{2\left(-56\right)}$$

Multiply 2 times -56.

$$x=\frac{-78±50}{-112}$$

Now solve the equation x=\frac{-78±50}{-112} when ± is plus. Add -78 to 50.

$$x=-\frac{28}{-112}$$

Reduce the fraction \frac{-28}{-112} to lowest terms by extracting and canceling out 28.

$$x=\frac{1}{4}$$

Now solve the equation x=\frac{-78±50}{-112} when ± is minus. Subtract 50 from -78.

$$x=-\frac{128}{-112}$$

Reduce the fraction \frac{-128}{-112} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

$$x=\frac{1}{4} x=\frac{8}{7}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{3},\frac{11}{16} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(16x-11\right), the least common multiple of 1-3x,16x-11.

$$\left(11-16x\right)\times 2x=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 11-16x by 2.

$$\left(22-32x\right)x=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 22-32x by x.

$$22x-32x^{2}=\left(3x-1\right)\left(8x-16\right)$$

Use the distributive property to multiply 3x-1 by 8x-16 and combine like terms.

$$22x-32x^{2}=24x^{2}-56x+16$$

Subtract 24x^{2} from both sides.

$$22x-32x^{2}-24x^{2}=-56x+16$$

Combine -32x^{2} and -24x^{2} to get -56x^{2}.

$$22x-56x^{2}=-56x+16$$

Add 56x to both sides.

$$22x-56x^{2}+56x=16$$

Combine 22x and 56x to get 78x.

$$78x-56x^{2}=16$$

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.

$$-56x^{2}+78x=16$$

Divide both sides by -56.

$$\frac{-56x^{2}+78x}{-56}=\frac{16}{-56}$$

Dividing by -56 undoes the multiplication by -56.

$$x^{2}+\frac{78}{-56}x=\frac{16}{-56}$$

Reduce the fraction \frac{78}{-56} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{39}{28}x=\frac{16}{-56}$$

Reduce the fraction \frac{16}{-56} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{39}{28}x=-\frac{2}{7}$$

Divide -\frac{39}{28}, the coefficient of the x term, by 2 to get -\frac{39}{56}. Then add the square of -\frac{39}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{39}{28}x+\left(-\frac{39}{56}\right)^{2}=-\frac{2}{7}+\left(-\frac{39}{56}\right)^{2}$$

Square -\frac{39}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{39}{28}x+\frac{1521}{3136}=-\frac{2}{7}+\frac{1521}{3136}$$

Add -\frac{2}{7} to \frac{1521}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{39}{28}x+\frac{1521}{3136}=\frac{625}{3136}$$

Factor x^{2}-\frac{39}{28}x+\frac{1521}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{56}\right)^{2}=\frac{625}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{56}\right)^{2}}=\sqrt{\frac{625}{3136}}$$

Simplify.

$$x-\frac{39}{56}=\frac{25}{56} x-\frac{39}{56}=-\frac{25}{56}$$

Add \frac{39}{56} to both sides of the equation.

$$x=\frac{8}{7} x=\frac{1}{4}$$