Solve (2x+1)(x-2)=(3x-2)(x-4)

$$( 2 x + 1 ) ( x - 2 ) = ( 3 x - 2 ) ( x - 4 )$$

Solve for x

$$x=1$$

$$x=10$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply 3x-2 by x-4 and combine like terms.

$$2x^{2}-3x-2=3x^{2}-14x+8$$

Subtract 3x^{2} from both sides.

$$2x^{2}-3x-2-3x^{2}=-14x+8$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-3x-2=-14x+8$$

Add 14x to both sides.

$$-x^{2}-3x-2+14x=8$$

Combine -3x and 14x to get 11x.

$$-x^{2}+11x-2=8$$

Subtract 8 from both sides.

$$-x^{2}+11x-2-8=0$$

Subtract 8 from -2 to get -10.

$$-x^{2}+11x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$x=\frac{-11±\sqrt{121-40}}{2\left(-1\right)}$$

Add 121 to -40.

$$x=\frac{-11±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-11±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±9}{-2}$$

Now solve the equation x=\frac{-11±9}{-2} when ± is plus. Add -11 to 9.

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

Divide -2 by -2.

$$x=1$$

Now solve the equation x=\frac{-11±9}{-2} when ± is minus. Subtract 9 from -11.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=1 x=10$$

Steps for Completing the Square

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

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

Use the distributive property to multiply 3x-2 by x-4 and combine like terms.

$$2x^{2}-3x-2=3x^{2}-14x+8$$

Subtract 3x^{2} from both sides.

$$2x^{2}-3x-2-3x^{2}=-14x+8$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-3x-2=-14x+8$$

Add 14x to both sides.

$$-x^{2}-3x-2+14x=8$$

Combine -3x and 14x to get 11x.

$$-x^{2}+11x-2=8$$

Add 2 to both sides.

$$-x^{2}+11x=8+2$$

Add 8 and 2 to get 10.

$$-x^{2}+11x=10$$

Divide both sides by -1.

$$\frac{-x^{2}+11x}{-1}=\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{11}{-1}x=\frac{10}{-1}$$

Divide 11 by -1.

$$x^{2}-11x=\frac{10}{-1}$$

Divide 10 by -1.

$$x^{2}-11x=-10$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-10+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-10+\frac{121}{4}$$

Add -10 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{81}{4}$$

Factor x^{2}-11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{9}{2} x-\frac{11}{2}=-\frac{9}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=10 x=1$$