Solve 4x-4+x=26x(2x-2)

$$4 x - 4 + x = 26 x ( 2 x - 2 )$$

Solve for x

$$x = \frac{\sqrt{2417} + 57}{104} \approx 1.020798021$$

$$x=\frac{57-\sqrt{2417}}{104}\approx 0.075355825$$

Steps Using the Quadratic Formula

Combine 4x and x to get 5x.

$$5x-4=26x\left(2x-2\right)$$

Use the distributive property to multiply 26x by 2x-2.

$$5x-4=52x^{2}-52x$$

Subtract 52x^{2} from both sides.

$$5x-4-52x^{2}=-52x$$

Add 52x to both sides.

$$5x-4-52x^{2}+52x=0$$

Combine 5x and 52x to get 57x.

$$57x-4-52x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-52x^{2}+57x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -52 for a, 57 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-57±\sqrt{57^{2}-4\left(-52\right)\left(-4\right)}}{2\left(-52\right)}$$

Square 57.

$$x=\frac{-57±\sqrt{3249-4\left(-52\right)\left(-4\right)}}{2\left(-52\right)}$$

Multiply -4 times -52.

$$x=\frac{-57±\sqrt{3249+208\left(-4\right)}}{2\left(-52\right)}$$

Multiply 208 times -4.

$$x=\frac{-57±\sqrt{3249-832}}{2\left(-52\right)}$$

Add 3249 to -832.

$$x=\frac{-57±\sqrt{2417}}{2\left(-52\right)}$$

Multiply 2 times -52.

$$x=\frac{-57±\sqrt{2417}}{-104}$$

Now solve the equation x=\frac{-57±\sqrt{2417}}{-104} when ± is plus. Add -57 to \sqrt{2417}.

$$x=\frac{\sqrt{2417}-57}{-104}$$

Divide -57+\sqrt{2417} by -104.

$$x=\frac{57-\sqrt{2417}}{104}$$

Now solve the equation x=\frac{-57±\sqrt{2417}}{-104} when ± is minus. Subtract \sqrt{2417} from -57.

$$x=\frac{-\sqrt{2417}-57}{-104}$$

Divide -57-\sqrt{2417} by -104.

$$x=\frac{\sqrt{2417}+57}{104}$$

The equation is now solved.

$$x=\frac{57-\sqrt{2417}}{104} x=\frac{\sqrt{2417}+57}{104}$$

Steps for Completing the Square

Combine 4x and x to get 5x.

$$5x-4=26x\left(2x-2\right)$$

Use the distributive property to multiply 26x by 2x-2.

$$5x-4=52x^{2}-52x$$

Subtract 52x^{2} from both sides.

$$5x-4-52x^{2}=-52x$$

Add 52x to both sides.

$$5x-4-52x^{2}+52x=0$$

Combine 5x and 52x to get 57x.

$$57x-4-52x^{2}=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$57x-52x^{2}=4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-52x^{2}+57x=4$$

Divide both sides by -52.

$$\frac{-52x^{2}+57x}{-52}=\frac{4}{-52}$$

Dividing by -52 undoes the multiplication by -52.

$$x^{2}+\frac{57}{-52}x=\frac{4}{-52}$$

Divide 57 by -52.

$$x^{2}-\frac{57}{52}x=\frac{4}{-52}$$

Reduce the fraction \frac{4}{-52} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{57}{52}x=-\frac{1}{13}$$

Divide -\frac{57}{52}, the coefficient of the x term, by 2 to get -\frac{57}{104}. Then add the square of -\frac{57}{104} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{57}{52}x+\left(-\frac{57}{104}\right)^{2}=-\frac{1}{13}+\left(-\frac{57}{104}\right)^{2}$$

Square -\frac{57}{104} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{57}{52}x+\frac{3249}{10816}=-\frac{1}{13}+\frac{3249}{10816}$$

Add -\frac{1}{13} to \frac{3249}{10816} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{57}{52}x+\frac{3249}{10816}=\frac{2417}{10816}$$

Factor x^{2}-\frac{57}{52}x+\frac{3249}{10816}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{57}{104}\right)^{2}=\frac{2417}{10816}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{57}{104}\right)^{2}}=\sqrt{\frac{2417}{10816}}$$

Simplify.

$$x-\frac{57}{104}=\frac{\sqrt{2417}}{104} x-\frac{57}{104}=-\frac{\sqrt{2417}}{104}$$

Add \frac{57}{104} to both sides of the equation.

$$x=\frac{\sqrt{2417}+57}{104} x=\frac{57-\sqrt{2417}}{104}$$