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

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

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

$$x=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x by x-4.

$$5x^{2}-20x=\left(2x+1\right)\left(x-4\right)$$

Use the distributive property to multiply 2x+1 by x-4 and combine like terms.

$$5x^{2}-20x=2x^{2}-7x-4$$

Subtract 2x^{2} from both sides.

$$5x^{2}-20x-2x^{2}=-7x-4$$

Combine 5x^{2} and -2x^{2} to get 3x^{2}.

$$3x^{2}-20x=-7x-4$$

Add 7x to both sides.

$$3x^{2}-20x+7x=-4$$

Combine -20x and 7x to get -13x.

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

Add 4 to both sides.

$$3x^{2}-13x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -13 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\times 4}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\times 4}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\times 4}}{2\times 3}$$

Multiply -12 times 4.

$$x=\frac{-\left(-13\right)±\sqrt{169-48}}{2\times 3}$$

Add 169 to -48.

$$x=\frac{-\left(-13\right)±\sqrt{121}}{2\times 3}$$

Take the square root of 121.

$$x=\frac{-\left(-13\right)±11}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±11}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±11}{6}$$

Now solve the equation x=\frac{13±11}{6} when ± is plus. Add 13 to 11.

$$x=\frac{24}{6}$$

Divide 24 by 6.

$$x=4$$

Now solve the equation x=\frac{13±11}{6} when ± is minus. Subtract 11 from 13.

$$x=\frac{2}{6}$$

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

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 5x by x-4.

$$5x^{2}-20x=\left(2x+1\right)\left(x-4\right)$$

Use the distributive property to multiply 2x+1 by x-4 and combine like terms.

$$5x^{2}-20x=2x^{2}-7x-4$$

Subtract 2x^{2} from both sides.

$$5x^{2}-20x-2x^{2}=-7x-4$$

Combine 5x^{2} and -2x^{2} to get 3x^{2}.

$$3x^{2}-20x=-7x-4$$

Add 7x to both sides.

$$3x^{2}-20x+7x=-4$$

Combine -20x and 7x to get -13x.

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

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

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.

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

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

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

Add -\frac{4}{3} to \frac{169}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=\frac{121}{36}$$

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}.

$$\left(x-\frac{13}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$x-\frac{13}{6}=\frac{11}{6} x-\frac{13}{6}=-\frac{11}{6}$$

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

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