Solve 6x^2-13x+4=4-2

$$6 x ^ { 2 } - 13 x + 4 = 4 - 2$$

Solve for x

$$x=\frac{1}{6}\approx 0.166666667$$

$$x=2$$

Steps Using Factoring By Grouping

Subtract 2 from 4 to get 2.

$$6x^{2}-13x+4=2$$

Subtract 2 from both sides.

$$6x^{2}-13x+4-2=0$$

Subtract 2 from 4 to get 2.

$$6x^{2}-13x+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-13 ab=6\times 2=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -13.

$$a=-12 b=-1$$

Rewrite 6x^{2}-13x+2 as \left(6x^{2}-12x\right)+\left(-x+2\right).

$$\left(6x^{2}-12x\right)+\left(-x+2\right)$$

Factor out 6x in the first and -1 in the second group.

$$6x\left(x-2\right)-\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(6x-1\right)$$

To find equation solutions, solve x-2=0 and 6x-1=0.

$$x=2 x=\frac{1}{6}$$

Steps Using the Quadratic Formula

Subtract 2 from 4 to get 2.

$$6x^{2}-13x+4=2$$

Subtract 2 from both sides.

$$6x^{2}-13x+4-2=0$$

Subtract 2 from 4 to get 2.

$$6x^{2}-13x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -13 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 2}}{2\times 6}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\times 2}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\times 2}}{2\times 6}$$

Multiply -24 times 2.

$$x=\frac{-\left(-13\right)±\sqrt{169-48}}{2\times 6}$$

Add 169 to -48.

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

Take the square root of 121.

$$x=\frac{-\left(-13\right)±11}{2\times 6}$$

The opposite of -13 is 13.

$$x=\frac{13±11}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{13±11}{12}$$

Now solve the equation x=\frac{13±11}{12} when ± is plus. Add 13 to 11.

$$x=\frac{24}{12}$$

Divide 24 by 12.

$$x=2$$

Now solve the equation x=\frac{13±11}{12} when ± is minus. Subtract 11 from 13.

$$x=\frac{2}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{6}$$

The equation is now solved.

$$x=2 x=\frac{1}{6}$$

Steps for Completing the Square

Subtract 2 from 4 to get 2.

$$6x^{2}-13x+4=2$$

Subtract 4 from both sides.

$$6x^{2}-13x=2-4$$

Subtract 4 from 2 to get -2.

$$6x^{2}-13x=-2$$

Divide both sides by 6.

$$\frac{6x^{2}-13x}{6}=-\frac{2}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{13}{6}x=-\frac{2}{6}$$

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

$$x^{2}-\frac{13}{6}x=-\frac{1}{3}$$

Divide -\frac{13}{6}, the coefficient of the x term, by 2 to get -\frac{13}{12}. Then add the square of -\frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-\frac{1}{3}+\left(-\frac{13}{12}\right)^{2}$$

Square -\frac{13}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-\frac{1}{3}+\frac{169}{144}$$

Add -\frac{1}{3} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{121}{144}$$

Factor x^{2}-\frac{13}{6}x+\frac{169}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{121}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{11}{12} x-\frac{13}{12}=-\frac{11}{12}$$

Add \frac{13}{12} to both sides of the equation.

$$x=2 x=\frac{1}{6}$$