Solve x-1/3=3x+4/6quad(9)quad2x

$$\frac { x - 1 } { 3 } = \frac { 3 x + 4 } { 6 } \quad ( 9 ) \quad 2 x$$

Solve for x

$$x=\frac{\sqrt{1117}-35}{54}\approx -0.029230557$$

$$x=\frac{-\sqrt{1117}-35}{54}\approx -1.26706574$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,6.

$$2\left(x-1\right)=\left(3x+4\right)\times 9\times 2x$$

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

$$2x-2=\left(3x+4\right)\times 9\times 2x$$

Multiply 9 and 2 to get 18.

$$2x-2=\left(3x+4\right)\times 18x$$

Use the distributive property to multiply 3x+4 by 18.

$$2x-2=\left(54x+72\right)x$$

Use the distributive property to multiply 54x+72 by x.

$$2x-2=54x^{2}+72x$$

Subtract 54x^{2} from both sides.

$$2x-2-54x^{2}=72x$$

Subtract 72x from both sides.

$$2x-2-54x^{2}-72x=0$$

Combine 2x and -72x to get -70x.

$$-70x-2-54x^{2}=0$$

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.

$$-54x^{2}-70x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -54 for a, -70 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\left(-54\right)\left(-2\right)}}{2\left(-54\right)}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\left(-54\right)\left(-2\right)}}{2\left(-54\right)}$$

Multiply -4 times -54.

$$x=\frac{-\left(-70\right)±\sqrt{4900+216\left(-2\right)}}{2\left(-54\right)}$$

Multiply 216 times -2.

$$x=\frac{-\left(-70\right)±\sqrt{4900-432}}{2\left(-54\right)}$$

Add 4900 to -432.

$$x=\frac{-\left(-70\right)±\sqrt{4468}}{2\left(-54\right)}$$

Take the square root of 4468.

$$x=\frac{-\left(-70\right)±2\sqrt{1117}}{2\left(-54\right)}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{1117}}{2\left(-54\right)}$$

Multiply 2 times -54.

$$x=\frac{70±2\sqrt{1117}}{-108}$$

Now solve the equation x=\frac{70±2\sqrt{1117}}{-108} when ± is plus. Add 70 to 2\sqrt{1117}.

$$x=\frac{2\sqrt{1117}+70}{-108}$$

Divide 70+2\sqrt{1117} by -108.

$$x=\frac{-\sqrt{1117}-35}{54}$$

Now solve the equation x=\frac{70±2\sqrt{1117}}{-108} when ± is minus. Subtract 2\sqrt{1117} from 70.

$$x=\frac{70-2\sqrt{1117}}{-108}$$

Divide 70-2\sqrt{1117} by -108.

$$x=\frac{\sqrt{1117}-35}{54}$$

The equation is now solved.

$$x=\frac{-\sqrt{1117}-35}{54} x=\frac{\sqrt{1117}-35}{54}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,6.

$$2\left(x-1\right)=\left(3x+4\right)\times 9\times 2x$$

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

$$2x-2=\left(3x+4\right)\times 9\times 2x$$

Multiply 9 and 2 to get 18.

$$2x-2=\left(3x+4\right)\times 18x$$

Use the distributive property to multiply 3x+4 by 18.

$$2x-2=\left(54x+72\right)x$$

Use the distributive property to multiply 54x+72 by x.

$$2x-2=54x^{2}+72x$$

Subtract 54x^{2} from both sides.

$$2x-2-54x^{2}=72x$$

Subtract 72x from both sides.

$$2x-2-54x^{2}-72x=0$$

Combine 2x and -72x to get -70x.

$$-70x-2-54x^{2}=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$-70x-54x^{2}=2$$

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.

$$-54x^{2}-70x=2$$

Divide both sides by -54.

$$\frac{-54x^{2}-70x}{-54}=\frac{2}{-54}$$

Dividing by -54 undoes the multiplication by -54.

$$x^{2}+\left(-\frac{70}{-54}\right)x=\frac{2}{-54}$$

Reduce the fraction \frac{-70}{-54} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{35}{27}x=\frac{2}{-54}$$

Reduce the fraction \frac{2}{-54} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{35}{27}x=-\frac{1}{27}$$

Divide \frac{35}{27}, the coefficient of the x term, by 2 to get \frac{35}{54}. Then add the square of \frac{35}{54} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{35}{27}x+\left(\frac{35}{54}\right)^{2}=-\frac{1}{27}+\left(\frac{35}{54}\right)^{2}$$

Square \frac{35}{54} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{35}{27}x+\frac{1225}{2916}=-\frac{1}{27}+\frac{1225}{2916}$$

Add -\frac{1}{27} to \frac{1225}{2916} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{35}{27}x+\frac{1225}{2916}=\frac{1117}{2916}$$

Factor x^{2}+\frac{35}{27}x+\frac{1225}{2916}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{35}{54}\right)^{2}=\frac{1117}{2916}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{35}{54}\right)^{2}}=\sqrt{\frac{1117}{2916}}$$

Simplify.

$$x+\frac{35}{54}=\frac{\sqrt{1117}}{54} x+\frac{35}{54}=-\frac{\sqrt{1117}}{54}$$

Subtract \frac{35}{54} from both sides of the equation.

$$x=\frac{\sqrt{1117}-35}{54} x=\frac{-\sqrt{1117}-35}{54}$$