Solve 5x(x-1)-4(x-1)=0

$$5 x ( x - 1 ) - 4 ( x - 1 ) = 0$$

Solve for x

$$x=\frac{4}{5}=0.8$$

$$x=1$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 5x by x-1.

$$5x^{2}-5x-4\left(x-1\right)=0$$

Use the distributive property to multiply -4 by x-1.

$$5x^{2}-5x-4x+4=0$$

Combine -5x and -4x to get -9x.

$$5x^{2}-9x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-9 ab=5\times 4=20$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -9.

$$a=-5 b=-4$$

Rewrite 5x^{2}-9x+4 as \left(5x^{2}-5x\right)+\left(-4x+4\right).

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

Factor out 5x in the first and -4 in the second group.

$$5x\left(x-1\right)-4\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(5x-4\right)$$

To find equation solutions, solve x-1=0 and 5x-4=0.

$$x=1 x=\frac{4}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x by x-1.

$$5x^{2}-5x-4\left(x-1\right)=0$$

Use the distributive property to multiply -4 by x-1.

$$5x^{2}-5x-4x+4=0$$

Combine -5x and -4x to get -9x.

$$5x^{2}-9x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -9 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 5\times 4}}{2\times 5}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 5\times 4}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-9\right)±\sqrt{81-20\times 4}}{2\times 5}$$

Multiply -20 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-80}}{2\times 5}$$

Add 81 to -80.

$$x=\frac{-\left(-9\right)±\sqrt{1}}{2\times 5}$$

Take the square root of 1.

$$x=\frac{-\left(-9\right)±1}{2\times 5}$$

The opposite of -9 is 9.

$$x=\frac{9±1}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{9±1}{10}$$

Now solve the equation x=\frac{9±1}{10} when ± is plus. Add 9 to 1.

$$x=\frac{10}{10}$$

Divide 10 by 10.

$$x=1$$

Now solve the equation x=\frac{9±1}{10} when ± is minus. Subtract 1 from 9.

$$x=\frac{8}{10}$$

Reduce the fraction \frac{8}{10} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=1 x=\frac{4}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 5x by x-1.

$$5x^{2}-5x-4\left(x-1\right)=0$$

Use the distributive property to multiply -4 by x-1.

$$5x^{2}-5x-4x+4=0$$

Combine -5x and -4x to get -9x.

$$5x^{2}-9x+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$5x^{2}-9x=-4$$

Divide both sides by 5.

$$\frac{5x^{2}-9x}{5}=-\frac{4}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{9}{5}x=-\frac{4}{5}$$

Divide -\frac{9}{5}, the coefficient of the x term, by 2 to get -\frac{9}{10}. Then add the square of -\frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=-\frac{4}{5}+\left(-\frac{9}{10}\right)^{2}$$

Square -\frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=-\frac{4}{5}+\frac{81}{100}$$

Add -\frac{4}{5} to \frac{81}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{1}{100}$$

Factor x^{2}-\frac{9}{5}x+\frac{81}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{10}\right)^{2}=\frac{1}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{\frac{1}{100}}$$

Simplify.

$$x-\frac{9}{10}=\frac{1}{10} x-\frac{9}{10}=-\frac{1}{10}$$

Add \frac{9}{10} to both sides of the equation.

$$x=1 x=\frac{4}{5}$$