Solve (6x-1)(2x+7)=(4-5x)(1-6x)

$$(6x-1)(2x+7)=(4-5x)(1-6x)$$

Solve for x

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

$$x = \frac{11}{3} = 3\frac{2}{3} \approx 3.666666667$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x-1 by 2x+7 and combine like terms.

$$12x^{2}+40x-7=\left(4-5x\right)\left(1-6x\right)$$

Use the distributive property to multiply 4-5x by 1-6x and combine like terms.

$$12x^{2}+40x-7=4-29x+30x^{2}$$

Subtract 4 from both sides.

$$12x^{2}+40x-7-4=-29x+30x^{2}$$

Subtract 4 from -7 to get -11.

$$12x^{2}+40x-11=-29x+30x^{2}$$

Add 29x to both sides.

$$12x^{2}+40x-11+29x=30x^{2}$$

Combine 40x and 29x to get 69x.

$$12x^{2}+69x-11=30x^{2}$$

Subtract 30x^{2} from both sides.

$$12x^{2}+69x-11-30x^{2}=0$$

Combine 12x^{2} and -30x^{2} to get -18x^{2}.

$$-18x^{2}+69x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 69 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-69±\sqrt{69^{2}-4\left(-18\right)\left(-11\right)}}{2\left(-18\right)}$$

Square 69.

$$x=\frac{-69±\sqrt{4761-4\left(-18\right)\left(-11\right)}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-69±\sqrt{4761+72\left(-11\right)}}{2\left(-18\right)}$$

Multiply 72 times -11.

$$x=\frac{-69±\sqrt{4761-792}}{2\left(-18\right)}$$

Add 4761 to -792.

$$x=\frac{-69±\sqrt{3969}}{2\left(-18\right)}$$

Take the square root of 3969.

$$x=\frac{-69±63}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-69±63}{-36}$$

Now solve the equation x=\frac{-69±63}{-36} when ± is plus. Add -69 to 63.

$$x=-\frac{6}{-36}$$

Reduce the fraction \frac{-6}{-36} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-69±63}{-36} when ± is minus. Subtract 63 from -69.

$$x=-\frac{132}{-36}$$

Reduce the fraction \frac{-132}{-36} to lowest terms by extracting and canceling out 12.

$$x=\frac{11}{3}$$

The equation is now solved.

$$x=\frac{1}{6} x=\frac{11}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 6x-1 by 2x+7 and combine like terms.

$$12x^{2}+40x-7=\left(4-5x\right)\left(1-6x\right)$$

Use the distributive property to multiply 4-5x by 1-6x and combine like terms.

$$12x^{2}+40x-7=4-29x+30x^{2}$$

Add 29x to both sides.

$$12x^{2}+40x-7+29x=4+30x^{2}$$

Combine 40x and 29x to get 69x.

$$12x^{2}+69x-7=4+30x^{2}$$

Subtract 30x^{2} from both sides.

$$12x^{2}+69x-7-30x^{2}=4$$

Combine 12x^{2} and -30x^{2} to get -18x^{2}.

$$-18x^{2}+69x-7=4$$

Add 7 to both sides.

$$-18x^{2}+69x=4+7$$

Add 4 and 7 to get 11.

$$-18x^{2}+69x=11$$

Divide both sides by -18.

$$\frac{-18x^{2}+69x}{-18}=\frac{11}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{69}{-18}x=\frac{11}{-18}$$

Reduce the fraction \frac{69}{-18} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{23}{6}x=\frac{11}{-18}$$

Divide 11 by -18.

$$x^{2}-\frac{23}{6}x=-\frac{11}{18}$$

Divide -\frac{23}{6}, the coefficient of the x term, by 2 to get -\frac{23}{12}. Then add the square of -\frac{23}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{6}x+\left(-\frac{23}{12}\right)^{2}=-\frac{11}{18}+\left(-\frac{23}{12}\right)^{2}$$

Square -\frac{23}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{6}x+\frac{529}{144}=-\frac{11}{18}+\frac{529}{144}$$

Add -\frac{11}{18} to \frac{529}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{6}x+\frac{529}{144}=\frac{49}{16}$$

Factor x^{2}-\frac{23}{6}x+\frac{529}{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}.

$$\left(x-\frac{23}{12}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{12}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{23}{12}=\frac{7}{4} x-\frac{23}{12}=-\frac{7}{4}$$

Add \frac{23}{12} to both sides of the equation.

$$x=\frac{11}{3} x=\frac{1}{6}$$