Solve 4left(x-3right)left(5x-19right)=4

$$4 \left( x-3 \right) \left( 5x-19 \right) = 4$$

Solve for x

$$x=4$$

$$x = \frac{14}{5} = 2\frac{4}{5} = 2.8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by x-3.

$$\left(4x-12\right)\left(5x-19\right)=4$$

Use the distributive property to multiply 4x-12 by 5x-19 and combine like terms.

$$20x^{2}-136x+228=4$$

Subtract 4 from both sides.

$$20x^{2}-136x+228-4=0$$

Subtract 4 from 228 to get 224.

$$20x^{2}-136x+224=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -136 for b, and 224 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-136\right)±\sqrt{\left(-136\right)^{2}-4\times 20\times 224}}{2\times 20}$$

Square -136.

$$x=\frac{-\left(-136\right)±\sqrt{18496-4\times 20\times 224}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-136\right)±\sqrt{18496-80\times 224}}{2\times 20}$$

Multiply -80 times 224.

$$x=\frac{-\left(-136\right)±\sqrt{18496-17920}}{2\times 20}$$

Add 18496 to -17920.

$$x=\frac{-\left(-136\right)±\sqrt{576}}{2\times 20}$$

Take the square root of 576.

$$x=\frac{-\left(-136\right)±24}{2\times 20}$$

The opposite of -136 is 136.

$$x=\frac{136±24}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{136±24}{40}$$

Now solve the equation x=\frac{136±24}{40} when ± is plus. Add 136 to 24.

$$x=\frac{160}{40}$$

Divide 160 by 40.

$$x=4$$

Now solve the equation x=\frac{136±24}{40} when ± is minus. Subtract 24 from 136.

$$x=\frac{112}{40}$$

Reduce the fraction \frac{112}{40} to lowest terms by extracting and canceling out 8.

$$x=\frac{14}{5}$$

The equation is now solved.

$$x=4 x=\frac{14}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 4 by x-3.

$$\left(4x-12\right)\left(5x-19\right)=4$$

Use the distributive property to multiply 4x-12 by 5x-19 and combine like terms.

$$20x^{2}-136x+228=4$$

Subtract 228 from both sides.

$$20x^{2}-136x=4-228$$

Subtract 228 from 4 to get -224.

$$20x^{2}-136x=-224$$

Divide both sides by 20.

$$\frac{20x^{2}-136x}{20}=-\frac{224}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}+\left(-\frac{136}{20}\right)x=-\frac{224}{20}$$

Reduce the fraction \frac{-136}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{34}{5}x=-\frac{224}{20}$$

Reduce the fraction \frac{-224}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{34}{5}x=-\frac{56}{5}$$

Divide -\frac{34}{5}, the coefficient of the x term, by 2 to get -\frac{17}{5}. Then add the square of -\frac{17}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{34}{5}x+\left(-\frac{17}{5}\right)^{2}=-\frac{56}{5}+\left(-\frac{17}{5}\right)^{2}$$

Square -\frac{17}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{34}{5}x+\frac{289}{25}=-\frac{56}{5}+\frac{289}{25}$$

Add -\frac{56}{5} to \frac{289}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{34}{5}x+\frac{289}{25}=\frac{9}{25}$$

Factor x^{2}-\frac{34}{5}x+\frac{289}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{5}\right)^{2}=\frac{9}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{5}\right)^{2}}=\sqrt{\frac{9}{25}}$$

Simplify.

$$x-\frac{17}{5}=\frac{3}{5} x-\frac{17}{5}=-\frac{3}{5}$$

Add \frac{17}{5} to both sides of the equation.

$$x=4 x=\frac{14}{5}$$