Solve 3/2x-1=5x-9

$$\frac { 3 } { 2 x - 1 } = 5 x - 9$$

Solve for x

$$x=2$$

$$x=\frac{3}{10}=0.3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-1.

$$3=5x\left(2x-1\right)+\left(2x-1\right)\left(-9\right)$$

Use the distributive property to multiply 5x by 2x-1.

$$3=10x^{2}-5x+\left(2x-1\right)\left(-9\right)$$

Use the distributive property to multiply 2x-1 by -9.

$$3=10x^{2}-5x-18x+9$$

Combine -5x and -18x to get -23x.

$$3=10x^{2}-23x+9$$

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

$$10x^{2}-23x+9=3$$

Subtract 3 from both sides.

$$10x^{2}-23x+9-3=0$$

Subtract 3 from 9 to get 6.

$$10x^{2}-23x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -23 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 10\times 6}}{2\times 10}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 10\times 6}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-23\right)±\sqrt{529-40\times 6}}{2\times 10}$$

Multiply -40 times 6.

$$x=\frac{-\left(-23\right)±\sqrt{529-240}}{2\times 10}$$

Add 529 to -240.

$$x=\frac{-\left(-23\right)±\sqrt{289}}{2\times 10}$$

Take the square root of 289.

$$x=\frac{-\left(-23\right)±17}{2\times 10}$$

The opposite of -23 is 23.

$$x=\frac{23±17}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{23±17}{20}$$

Now solve the equation x=\frac{23±17}{20} when ± is plus. Add 23 to 17.

$$x=\frac{40}{20}$$

Divide 40 by 20.

$$x=2$$

Now solve the equation x=\frac{23±17}{20} when ± is minus. Subtract 17 from 23.

$$x=\frac{6}{20}$$

Reduce the fraction \frac{6}{20} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{10}$$

The equation is now solved.

$$x=2 x=\frac{3}{10}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-1.

$$3=5x\left(2x-1\right)+\left(2x-1\right)\left(-9\right)$$

Use the distributive property to multiply 5x by 2x-1.

$$3=10x^{2}-5x+\left(2x-1\right)\left(-9\right)$$

Use the distributive property to multiply 2x-1 by -9.

$$3=10x^{2}-5x-18x+9$$

Combine -5x and -18x to get -23x.

$$3=10x^{2}-23x+9$$

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

$$10x^{2}-23x+9=3$$

Subtract 9 from both sides.

$$10x^{2}-23x=3-9$$

Subtract 9 from 3 to get -6.

$$10x^{2}-23x=-6$$

Divide both sides by 10.

$$\frac{10x^{2}-23x}{10}=-\frac{6}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}-\frac{23}{10}x=-\frac{6}{10}$$

Reduce the fraction \frac{-6}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{23}{10}x=-\frac{3}{5}$$

Divide -\frac{23}{10}, the coefficient of the x term, by 2 to get -\frac{23}{20}. Then add the square of -\frac{23}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{10}x+\left(-\frac{23}{20}\right)^{2}=-\frac{3}{5}+\left(-\frac{23}{20}\right)^{2}$$

Square -\frac{23}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{10}x+\frac{529}{400}=-\frac{3}{5}+\frac{529}{400}$$

Add -\frac{3}{5} to \frac{529}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{10}x+\frac{529}{400}=\frac{289}{400}$$

Factor x^{2}-\frac{23}{10}x+\frac{529}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{20}\right)^{2}=\frac{289}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{20}\right)^{2}}=\sqrt{\frac{289}{400}}$$

Simplify.

$$x-\frac{23}{20}=\frac{17}{20} x-\frac{23}{20}=-\frac{17}{20}$$

Add \frac{23}{20} to both sides of the equation.

$$x=2 x=\frac{3}{10}$$