Solve (2x-1)(3-6x)=(3x-5)(10-6x)

$$( 2 x - 1 ) ( 3 - 6 x ) = ( 3 x - 5 ) ( 10 - 6 x )$$

Solve for x

$$x=\frac{7\sqrt{6}}{6}+4\approx 6.857738033$$

$$x=-\frac{7\sqrt{6}}{6}+4\approx 1.142261967$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-1 by 3-6x and combine like terms.

$$12x-12x^{2}-3=\left(3x-5\right)\left(10-6x\right)$$

Use the distributive property to multiply 3x-5 by 10-6x and combine like terms.

$$12x-12x^{2}-3=60x-18x^{2}-50$$

Subtract 60x from both sides.

$$12x-12x^{2}-3-60x=-18x^{2}-50$$

Combine 12x and -60x to get -48x.

$$-48x-12x^{2}-3=-18x^{2}-50$$

Add 18x^{2} to both sides.

$$-48x-12x^{2}-3+18x^{2}=-50$$

Combine -12x^{2} and 18x^{2} to get 6x^{2}.

$$-48x+6x^{2}-3=-50$$

Add 50 to both sides.

$$-48x+6x^{2}-3+50=0$$

Add -3 and 50 to get 47.

$$-48x+6x^{2}+47=0$$

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.

$$6x^{2}-48x+47=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -48 for b, and 47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 6\times 47}}{2\times 6}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 6\times 47}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-48\right)±\sqrt{2304-24\times 47}}{2\times 6}$$

Multiply -24 times 47.

$$x=\frac{-\left(-48\right)±\sqrt{2304-1128}}{2\times 6}$$

Add 2304 to -1128.

$$x=\frac{-\left(-48\right)±\sqrt{1176}}{2\times 6}$$

Take the square root of 1176.

$$x=\frac{-\left(-48\right)±14\sqrt{6}}{2\times 6}$$

The opposite of -48 is 48.

$$x=\frac{48±14\sqrt{6}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{48±14\sqrt{6}}{12}$$

Now solve the equation x=\frac{48±14\sqrt{6}}{12} when ± is plus. Add 48 to 14\sqrt{6}.

$$x=\frac{14\sqrt{6}+48}{12}$$

Divide 48+14\sqrt{6} by 12.

$$x=\frac{7\sqrt{6}}{6}+4$$

Now solve the equation x=\frac{48±14\sqrt{6}}{12} when ± is minus. Subtract 14\sqrt{6} from 48.

$$x=\frac{48-14\sqrt{6}}{12}$$

Divide 48-14\sqrt{6} by 12.

$$x=-\frac{7\sqrt{6}}{6}+4$$

The equation is now solved.

$$x=\frac{7\sqrt{6}}{6}+4 x=-\frac{7\sqrt{6}}{6}+4$$

Steps for Completing the Square

Use the distributive property to multiply 2x-1 by 3-6x and combine like terms.

$$12x-12x^{2}-3=\left(3x-5\right)\left(10-6x\right)$$

Use the distributive property to multiply 3x-5 by 10-6x and combine like terms.

$$12x-12x^{2}-3=60x-18x^{2}-50$$

Subtract 60x from both sides.

$$12x-12x^{2}-3-60x=-18x^{2}-50$$

Combine 12x and -60x to get -48x.

$$-48x-12x^{2}-3=-18x^{2}-50$$

Add 18x^{2} to both sides.

$$-48x-12x^{2}-3+18x^{2}=-50$$

Combine -12x^{2} and 18x^{2} to get 6x^{2}.

$$-48x+6x^{2}-3=-50$$

Add 3 to both sides.

$$-48x+6x^{2}=-50+3$$

Add -50 and 3 to get -47.

$$-48x+6x^{2}=-47$$

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.

$$6x^{2}-48x=-47$$

Divide both sides by 6.

$$\frac{6x^{2}-48x}{6}=-\frac{47}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{48}{6}\right)x=-\frac{47}{6}$$

Divide -48 by 6.

$$x^{2}-8x=-\frac{47}{6}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{47}{6}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-\frac{47}{6}+16$$

Add -\frac{47}{6} to 16.

$$x^{2}-8x+16=\frac{49}{6}$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=\frac{49}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{\frac{49}{6}}$$

Simplify.

$$x-4=\frac{7\sqrt{6}}{6} x-4=-\frac{7\sqrt{6}}{6}$$

Add 4 to both sides of the equation.

$$x=\frac{7\sqrt{6}}{6}+4 x=-\frac{7\sqrt{6}}{6}+4$$