Solve (6x-5)^2=0

$$( 6 x - 5 ) ^ { 2 } = 0$$

Solve for x

$$x=\frac{5}{6}\approx 0.833333333$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-60x+25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 36x^{2}+ax+bx+25. To find a and b, set up a system to be solved.

$$a+b=-60 ab=36\times 25=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Calculate the sum for each pair.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

The solution is the pair that gives sum -60.

$$a=-30 b=-30$$

Rewrite 36x^{2}-60x+25 as \left(36x^{2}-30x\right)+\left(-30x+25\right).

$$\left(36x^{2}-30x\right)+\left(-30x+25\right)$$

Factor out 6x in the first and -5 in the second group.

$$6x\left(6x-5\right)-5\left(6x-5\right)$$

Factor out common term 6x-5 by using distributive property.

$$\left(6x-5\right)\left(6x-5\right)$$

Rewrite as a binomial square.

$$\left(6x-5\right)^{2}$$

To find equation solution, solve 6x-5=0.

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

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-60x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -60 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 36\times 25}}{2\times 36}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 36\times 25}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-60\right)±\sqrt{3600-144\times 25}}{2\times 36}$$

Multiply -144 times 25.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3600}}{2\times 36}$$

Add 3600 to -3600.

$$x=\frac{-\left(-60\right)±\sqrt{0}}{2\times 36}$$

Take the square root of 0.

$$x=-\frac{-60}{2\times 36}$$

The opposite of -60 is 60.

$$x=\frac{60}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{60}{72}$$

Reduce the fraction \frac{60}{72} to lowest terms by extracting and canceling out 12.

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

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-60x+25=0$$

Subtract 25 from both sides. Anything subtracted from zero gives its negation.

$$36x^{2}-60x=-25$$

Divide both sides by 36.

$$\frac{36x^{2}-60x}{36}=-\frac{25}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{60}{36}\right)x=-\frac{25}{36}$$

Reduce the fraction \frac{-60}{36} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{5}{3}x=-\frac{25}{36}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=-\frac{25}{36}+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{-25+25}{36}$$

Add -\frac{25}{36} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=0$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{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}.

$$\left(x-\frac{5}{6}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{5}{6}=0 x-\frac{5}{6}=0$$

Add \frac{5}{6} to both sides of the equation.

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

The equation is now solved. Solutions are the same.

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