Solve (x-1)^2/5-x+4/6=2(x-1)/3

$$\frac { ( x - 1 ) ^ { 2 } } { 5 } - \frac { x + 4 } { 6 } = \frac { 2 ( x - 1 ) } { 3 }$$

Solve for x

$$x=\frac{1}{6}\approx 0.166666667$$

$$x=6$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 30, the least common multiple of 5,6,3.

$$6\left(x-1\right)^{2}-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$6\left(x^{2}-2x+1\right)-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply 6 by x^{2}-2x+1.

$$6x^{2}-12x+6-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply -5 by x+4.

$$6x^{2}-12x+6-5x-20=10\times 2\left(x-1\right)$$

Combine -12x and -5x to get -17x.

$$6x^{2}-17x+6-20=10\times 2\left(x-1\right)$$

Subtract 20 from 6 to get -14.

$$6x^{2}-17x-14=10\times 2\left(x-1\right)$$

Multiply 10 and 2 to get 20.

$$6x^{2}-17x-14=20\left(x-1\right)$$

Use the distributive property to multiply 20 by x-1.

$$6x^{2}-17x-14=20x-20$$

Subtract 20x from both sides.

$$6x^{2}-17x-14-20x=-20$$

Combine -17x and -20x to get -37x.

$$6x^{2}-37x-14=-20$$

Add 20 to both sides.

$$6x^{2}-37x-14+20=0$$

Add -14 and 20 to get 6.

$$6x^{2}-37x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-37 ab=6\times 6=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -37.

$$a=-36 b=-1$$

Rewrite 6x^{2}-37x+6 as \left(6x^{2}-36x\right)+\left(-x+6\right).

$$\left(6x^{2}-36x\right)+\left(-x+6\right)$$

Factor out 6x in the first and -1 in the second group.

$$6x\left(x-6\right)-\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(6x-1\right)$$

To find equation solutions, solve x-6=0 and 6x-1=0.

$$x=6 x=\frac{1}{6}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 5,6,3.

$$6\left(x-1\right)^{2}-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$6\left(x^{2}-2x+1\right)-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply 6 by x^{2}-2x+1.

$$6x^{2}-12x+6-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply -5 by x+4.

$$6x^{2}-12x+6-5x-20=10\times 2\left(x-1\right)$$

Combine -12x and -5x to get -17x.

$$6x^{2}-17x+6-20=10\times 2\left(x-1\right)$$

Subtract 20 from 6 to get -14.

$$6x^{2}-17x-14=10\times 2\left(x-1\right)$$

Multiply 10 and 2 to get 20.

$$6x^{2}-17x-14=20\left(x-1\right)$$

Use the distributive property to multiply 20 by x-1.

$$6x^{2}-17x-14=20x-20$$

Subtract 20x from both sides.

$$6x^{2}-17x-14-20x=-20$$

Combine -17x and -20x to get -37x.

$$6x^{2}-37x-14=-20$$

Add 20 to both sides.

$$6x^{2}-37x-14+20=0$$

Add -14 and 20 to get 6.

$$6x^{2}-37x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -37 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 6\times 6}}{2\times 6}$$

Square -37.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\times 6\times 6}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-37\right)±\sqrt{1369-24\times 6}}{2\times 6}$$

Multiply -24 times 6.

$$x=\frac{-\left(-37\right)±\sqrt{1369-144}}{2\times 6}$$

Add 1369 to -144.

$$x=\frac{-\left(-37\right)±\sqrt{1225}}{2\times 6}$$

Take the square root of 1225.

$$x=\frac{-\left(-37\right)±35}{2\times 6}$$

The opposite of -37 is 37.

$$x=\frac{37±35}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{37±35}{12}$$

Now solve the equation x=\frac{37±35}{12} when ± is plus. Add 37 to 35.

$$x=\frac{72}{12}$$

Divide 72 by 12.

$$x=6$$

Now solve the equation x=\frac{37±35}{12} when ± is minus. Subtract 35 from 37.

$$x=\frac{2}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{6}$$

The equation is now solved.

$$x=6 x=\frac{1}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 5,6,3.

$$6\left(x-1\right)^{2}-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$6\left(x^{2}-2x+1\right)-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply 6 by x^{2}-2x+1.

$$6x^{2}-12x+6-5\left(x+4\right)=10\times 2\left(x-1\right)$$

Use the distributive property to multiply -5 by x+4.

$$6x^{2}-12x+6-5x-20=10\times 2\left(x-1\right)$$

Combine -12x and -5x to get -17x.

$$6x^{2}-17x+6-20=10\times 2\left(x-1\right)$$

Subtract 20 from 6 to get -14.

$$6x^{2}-17x-14=10\times 2\left(x-1\right)$$

Multiply 10 and 2 to get 20.

$$6x^{2}-17x-14=20\left(x-1\right)$$

Use the distributive property to multiply 20 by x-1.

$$6x^{2}-17x-14=20x-20$$

Subtract 20x from both sides.

$$6x^{2}-17x-14-20x=-20$$

Combine -17x and -20x to get -37x.

$$6x^{2}-37x-14=-20$$

Add 14 to both sides.

$$6x^{2}-37x=-20+14$$

Add -20 and 14 to get -6.

$$6x^{2}-37x=-6$$

Divide both sides by 6.

$$\frac{6x^{2}-37x}{6}=-\frac{6}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{37}{6}x=-\frac{6}{6}$$

Divide -6 by 6.

$$x^{2}-\frac{37}{6}x=-1$$

Divide -\frac{37}{6}, the coefficient of the x term, by 2 to get -\frac{37}{12}. Then add the square of -\frac{37}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{37}{6}x+\left(-\frac{37}{12}\right)^{2}=-1+\left(-\frac{37}{12}\right)^{2}$$

Square -\frac{37}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{37}{6}x+\frac{1369}{144}=-1+\frac{1369}{144}$$

Add -1 to \frac{1369}{144}.

$$x^{2}-\frac{37}{6}x+\frac{1369}{144}=\frac{1225}{144}$$

Factor x^{2}-\frac{37}{6}x+\frac{1369}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{12}\right)^{2}=\frac{1225}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{12}\right)^{2}}=\sqrt{\frac{1225}{144}}$$

Simplify.

$$x-\frac{37}{12}=\frac{35}{12} x-\frac{37}{12}=-\frac{35}{12}$$

Add \frac{37}{12} to both sides of the equation.

$$x=6 x=\frac{1}{6}$$