Solve 4left(2x-13right)^2-9(2x-13)+2=0

$$4 { \left(2x-13 \right) }^{ 2 } -9(2x-13)+2=0$$

Solve for x

$$x = \frac{53}{8} = 6\frac{5}{8} = 6.625$$

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-13\right)^{2}.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

Use the distributive property to multiply 4 by 4x^{2}-52x+169.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

Use the distributive property to multiply -9 by 2x-13.

$$16x^{2}-208x+676-18x+117+2=0$$

Combine -208x and -18x to get -226x.

$$16x^{2}-226x+676+117+2=0$$

Add 676 and 117 to get 793.

$$16x^{2}-226x+793+2=0$$

Add 793 and 2 to get 795.

$$16x^{2}-226x+795=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -226 for b, and 795 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-226\right)±\sqrt{\left(-226\right)^{2}-4\times 16\times 795}}{2\times 16}$$

Square -226.

$$x=\frac{-\left(-226\right)±\sqrt{51076-4\times 16\times 795}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-226\right)±\sqrt{51076-64\times 795}}{2\times 16}$$

Multiply -64 times 795.

$$x=\frac{-\left(-226\right)±\sqrt{51076-50880}}{2\times 16}$$

Add 51076 to -50880.

$$x=\frac{-\left(-226\right)±\sqrt{196}}{2\times 16}$$

Take the square root of 196.

$$x=\frac{-\left(-226\right)±14}{2\times 16}$$

The opposite of -226 is 226.

$$x=\frac{226±14}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{226±14}{32}$$

Now solve the equation x=\frac{226±14}{32} when ± is plus. Add 226 to 14.

$$x=\frac{240}{32}$$

Reduce the fraction \frac{240}{32} to lowest terms by extracting and canceling out 16.

$$x=\frac{15}{2}$$

Now solve the equation x=\frac{226±14}{32} when ± is minus. Subtract 14 from 226.

$$x=\frac{212}{32}$$

Reduce the fraction \frac{212}{32} to lowest terms by extracting and canceling out 4.

$$x=\frac{53}{8}$$

The equation is now solved.

$$x=\frac{15}{2} x=\frac{53}{8}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-13\right)^{2}.

$$4\left(4x^{2}-52x+169\right)-9\left(2x-13\right)+2=0$$

Use the distributive property to multiply 4 by 4x^{2}-52x+169.

$$16x^{2}-208x+676-9\left(2x-13\right)+2=0$$

Use the distributive property to multiply -9 by 2x-13.

$$16x^{2}-208x+676-18x+117+2=0$$

Combine -208x and -18x to get -226x.

$$16x^{2}-226x+676+117+2=0$$

Add 676 and 117 to get 793.

$$16x^{2}-226x+793+2=0$$

Add 793 and 2 to get 795.

$$16x^{2}-226x+795=0$$

Subtract 795 from both sides. Anything subtracted from zero gives its negation.

$$16x^{2}-226x=-795$$

Divide both sides by 16.

$$\frac{16x^{2}-226x}{16}=-\frac{795}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{226}{16}\right)x=-\frac{795}{16}$$

Reduce the fraction \frac{-226}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{113}{8}x=-\frac{795}{16}$$

Divide -\frac{113}{8}, the coefficient of the x term, by 2 to get -\frac{113}{16}. Then add the square of -\frac{113}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{113}{8}x+\left(-\frac{113}{16}\right)^{2}=-\frac{795}{16}+\left(-\frac{113}{16}\right)^{2}$$

Square -\frac{113}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=-\frac{795}{16}+\frac{12769}{256}$$

Add -\frac{795}{16} to \frac{12769}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{113}{8}x+\frac{12769}{256}=\frac{49}{256}$$

Factor x^{2}-\frac{113}{8}x+\frac{12769}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{113}{16}\right)^{2}=\frac{49}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{113}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Simplify.

$$x-\frac{113}{16}=\frac{7}{16} x-\frac{113}{16}=-\frac{7}{16}$$

Add \frac{113}{16} to both sides of the equation.

$$x=\frac{15}{2} x=\frac{53}{8}$$