Solve (x-3)(2x-5)=(x-3)(3-4x)

$$( x - 3 ) ( 2 x - 5 ) = ( x - 3 ) ( 3 - 4 x )$$

Solve for x

$$x=3$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-3 by 2x-5 and combine like terms.

$$2x^{2}-11x+15=\left(x-3\right)\left(3-4x\right)$$

Use the distributive property to multiply x-3 by 3-4x and combine like terms.

$$2x^{2}-11x+15=15x-4x^{2}-9$$

Subtract 15x from both sides.

$$2x^{2}-11x+15-15x=-4x^{2}-9$$

Combine -11x and -15x to get -26x.

$$2x^{2}-26x+15=-4x^{2}-9$$

Add 4x^{2} to both sides.

$$2x^{2}-26x+15+4x^{2}=-9$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-26x+15=-9$$

Add 9 to both sides.

$$6x^{2}-26x+15+9=0$$

Add 15 and 9 to get 24.

$$6x^{2}-26x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -26 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 6\times 24}}{2\times 6}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 6\times 24}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-26\right)±\sqrt{676-24\times 24}}{2\times 6}$$

Multiply -24 times 24.

$$x=\frac{-\left(-26\right)±\sqrt{676-576}}{2\times 6}$$

Add 676 to -576.

$$x=\frac{-\left(-26\right)±\sqrt{100}}{2\times 6}$$

Take the square root of 100.

$$x=\frac{-\left(-26\right)±10}{2\times 6}$$

The opposite of -26 is 26.

$$x=\frac{26±10}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{26±10}{12}$$

Now solve the equation x=\frac{26±10}{12} when ± is plus. Add 26 to 10.

$$x=\frac{36}{12}$$

Divide 36 by 12.

$$x=3$$

Now solve the equation x=\frac{26±10}{12} when ± is minus. Subtract 10 from 26.

$$x=\frac{16}{12}$$

Reduce the fraction \frac{16}{12} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply x-3 by 2x-5 and combine like terms.

$$2x^{2}-11x+15=\left(x-3\right)\left(3-4x\right)$$

Use the distributive property to multiply x-3 by 3-4x and combine like terms.

$$2x^{2}-11x+15=15x-4x^{2}-9$$

Subtract 15x from both sides.

$$2x^{2}-11x+15-15x=-4x^{2}-9$$

Combine -11x and -15x to get -26x.

$$2x^{2}-26x+15=-4x^{2}-9$$

Add 4x^{2} to both sides.

$$2x^{2}-26x+15+4x^{2}=-9$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-26x+15=-9$$

Subtract 15 from both sides.

$$6x^{2}-26x=-9-15$$

Subtract 15 from -9 to get -24.

$$6x^{2}-26x=-24$$

Divide both sides by 6.

$$\frac{6x^{2}-26x}{6}=-\frac{24}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{26}{6}\right)x=-\frac{24}{6}$$

Reduce the fraction \frac{-26}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{13}{3}x=-\frac{24}{6}$$

Divide -24 by 6.

$$x^{2}-\frac{13}{3}x=-4$$

Divide -\frac{13}{3}, the coefficient of the x term, by 2 to get -\frac{13}{6}. Then add the square of -\frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{3}x+\left(-\frac{13}{6}\right)^{2}=-4+\left(-\frac{13}{6}\right)^{2}$$

Square -\frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=-4+\frac{169}{36}$$

Add -4 to \frac{169}{36}.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=\frac{25}{36}$$

Factor x^{2}-\frac{13}{3}x+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{6}\right)^{2}=\frac{25}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{6}\right)^{2}}=\sqrt{\frac{25}{36}}$$

Simplify.

$$x-\frac{13}{6}=\frac{5}{6} x-\frac{13}{6}=-\frac{5}{6}$$

Add \frac{13}{6} to both sides of the equation.

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