Solve 6(2x-1/6)^2-11(2x-1/6)-35=0

$$6 ( 2 x - \frac { 1 } { 6 } ) ^ { 2 } - 11 ( 2 x - \frac { 1 } { 6 } ) - 35 = 0$$

Solve for x

$$x=-\frac{3}{4}=-0.75$$

$$x = \frac{11}{6} = 1\frac{5}{6} \approx 1.833333333$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-\frac{1}{6}\right)^{2}.

$$6\left(4x^{2}-\frac{2}{3}x+\frac{1}{36}\right)-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply 6 by 4x^{2}-\frac{2}{3}x+\frac{1}{36}.

$$24x^{2}-4x+\frac{1}{6}-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply -11 by 2x-\frac{1}{6}.

$$24x^{2}-4x+\frac{1}{6}-22x+\frac{11}{6}-35=0$$

Combine -4x and -22x to get -26x.

$$24x^{2}-26x+\frac{1}{6}+\frac{11}{6}-35=0$$

Add \frac{1}{6} and \frac{11}{6} to get 2.

$$24x^{2}-26x+2-35=0$$

Subtract 35 from 2 to get -33.

$$24x^{2}-26x-33=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 24x^{2}+ax+bx-33. To find a and b, set up a system to be solved.

$$a+b=-26 ab=24\left(-33\right)=-792$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -792.

$$1,-792 2,-396 3,-264 4,-198 6,-132 8,-99 9,-88 11,-72 12,-66 18,-44 22,-36 24,-33$$

Calculate the sum for each pair.

$$1-792=-791 2-396=-394 3-264=-261 4-198=-194 6-132=-126 8-99=-91 9-88=-79 11-72=-61 12-66=-54 18-44=-26 22-36=-14 24-33=-9$$

The solution is the pair that gives sum -26.

$$a=-44 b=18$$

Rewrite 24x^{2}-26x-33 as \left(24x^{2}-44x\right)+\left(18x-33\right).

$$\left(24x^{2}-44x\right)+\left(18x-33\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(6x-11\right)+3\left(6x-11\right)$$

Factor out common term 6x-11 by using distributive property.

$$\left(6x-11\right)\left(4x+3\right)$$

To find equation solutions, solve 6x-11=0 and 4x+3=0.

$$x=\frac{11}{6} x=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-\frac{1}{6}\right)^{2}.

$$6\left(4x^{2}-\frac{2}{3}x+\frac{1}{36}\right)-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply 6 by 4x^{2}-\frac{2}{3}x+\frac{1}{36}.

$$24x^{2}-4x+\frac{1}{6}-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply -11 by 2x-\frac{1}{6}.

$$24x^{2}-4x+\frac{1}{6}-22x+\frac{11}{6}-35=0$$

Combine -4x and -22x to get -26x.

$$24x^{2}-26x+\frac{1}{6}+\frac{11}{6}-35=0$$

Add \frac{1}{6} and \frac{11}{6} to get 2.

$$24x^{2}-26x+2-35=0$$

Subtract 35 from 2 to get -33.

$$24x^{2}-26x-33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -26 for b, and -33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 24\left(-33\right)}}{2\times 24}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 24\left(-33\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-26\right)±\sqrt{676-96\left(-33\right)}}{2\times 24}$$

Multiply -96 times -33.

$$x=\frac{-\left(-26\right)±\sqrt{676+3168}}{2\times 24}$$

Add 676 to 3168.

$$x=\frac{-\left(-26\right)±\sqrt{3844}}{2\times 24}$$

Take the square root of 3844.

$$x=\frac{-\left(-26\right)±62}{2\times 24}$$

The opposite of -26 is 26.

$$x=\frac{26±62}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{26±62}{48}$$

Now solve the equation x=\frac{26±62}{48} when ± is plus. Add 26 to 62.

$$x=\frac{88}{48}$$

Reduce the fraction \frac{88}{48} to lowest terms by extracting and canceling out 8.

$$x=\frac{11}{6}$$

Now solve the equation x=\frac{26±62}{48} when ± is minus. Subtract 62 from 26.

$$x=-\frac{36}{48}$$

Reduce the fraction \frac{-36}{48} to lowest terms by extracting and canceling out 12.

$$x=-\frac{3}{4}$$

The equation is now solved.

$$x=\frac{11}{6} x=-\frac{3}{4}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-\frac{1}{6}\right)^{2}.

$$6\left(4x^{2}-\frac{2}{3}x+\frac{1}{36}\right)-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply 6 by 4x^{2}-\frac{2}{3}x+\frac{1}{36}.

$$24x^{2}-4x+\frac{1}{6}-11\left(2x-\frac{1}{6}\right)-35=0$$

Use the distributive property to multiply -11 by 2x-\frac{1}{6}.

$$24x^{2}-4x+\frac{1}{6}-22x+\frac{11}{6}-35=0$$

Combine -4x and -22x to get -26x.

$$24x^{2}-26x+\frac{1}{6}+\frac{11}{6}-35=0$$

Add \frac{1}{6} and \frac{11}{6} to get 2.

$$24x^{2}-26x+2-35=0$$

Subtract 35 from 2 to get -33.

$$24x^{2}-26x-33=0$$

Add 33 to both sides. Anything plus zero gives itself.

$$24x^{2}-26x=33$$

Divide both sides by 24.

$$\frac{24x^{2}-26x}{24}=\frac{33}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{26}{24}\right)x=\frac{33}{24}$$

Reduce the fraction \frac{-26}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{12}x=\frac{33}{24}$$

Reduce the fraction \frac{33}{24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{13}{12}x=\frac{11}{8}$$

Divide -\frac{13}{12}, the coefficient of the x term, by 2 to get -\frac{13}{24}. Then add the square of -\frac{13}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{12}x+\left(-\frac{13}{24}\right)^{2}=\frac{11}{8}+\left(-\frac{13}{24}\right)^{2}$$

Square -\frac{13}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{11}{8}+\frac{169}{576}$$

Add \frac{11}{8} to \frac{169}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{12}x+\frac{169}{576}=\frac{961}{576}$$

Factor x^{2}-\frac{13}{12}x+\frac{169}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{24}\right)^{2}=\frac{961}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{24}\right)^{2}}=\sqrt{\frac{961}{576}}$$

Simplify.

$$x-\frac{13}{24}=\frac{31}{24} x-\frac{13}{24}=-\frac{31}{24}$$

Add \frac{13}{24} to both sides of the equation.

$$x=\frac{11}{6} x=-\frac{3}{4}$$