Solve (17x-1)^2/13^2=(x-1)(2x)

$$\frac { ( 17 x - 1 ) ^ { 2 } } { 13 ^ { 2 } } = ( x - 1 ) ( 2 x )$$

Solve for x

$$x = \frac{13 \sqrt{137} + 152}{49} \approx 6.207369364$$

$$x=\frac{152-13\sqrt{137}}{49}\approx -0.003287731$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(17x-1\right)^{2}.

$$\frac{289x^{2}-34x+1}{13^{2}}=\left(x-1\right)\times 2x$$

Calculate 13 to the power of 2 and get 169.

$$\frac{289x^{2}-34x+1}{169}=\left(x-1\right)\times 2x$$

Use the distributive property to multiply x-1 by 2.

$$\frac{289x^{2}-34x+1}{169}=\left(2x-2\right)x$$

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

$$\frac{289x^{2}-34x+1}{169}=2x^{2}-2x$$

Divide each term of 289x^{2}-34x+1 by 169 to get \frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=2x^{2}-2x$$

Subtract 2x^{2} from both sides.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}-2x^{2}=-2x$$

Combine \frac{289}{169}x^{2} and -2x^{2} to get -\frac{49}{169}x^{2}.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=-2x$$

Add 2x to both sides.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}+2x=0$$

Combine -\frac{34}{169}x and 2x to get \frac{304}{169}x.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x+\frac{1}{169}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{49}{169} for a, \frac{304}{169} for b, and \frac{1}{169} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{304}{169}±\sqrt{\left(\frac{304}{169}\right)^{2}-4\left(-\frac{49}{169}\right)\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Square \frac{304}{169} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416}{28561}-4\left(-\frac{49}{169}\right)\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Multiply -4 times -\frac{49}{169}.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416}{28561}+\frac{196}{169}\times \frac{1}{169}}}{2\left(-\frac{49}{169}\right)}$$

Multiply \frac{196}{169} times \frac{1}{169} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{92416+196}{28561}}}{2\left(-\frac{49}{169}\right)}$$

Add \frac{92416}{28561} to \frac{196}{28561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{304}{169}±\sqrt{\frac{548}{169}}}{2\left(-\frac{49}{169}\right)}$$

Take the square root of \frac{548}{169}.

$$x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{2\left(-\frac{49}{169}\right)}$$

Multiply 2 times -\frac{49}{169}.

$$x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}}$$

Now solve the equation x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}} when ± is plus. Add -\frac{304}{169} to \frac{2\sqrt{137}}{13}.

$$x=\frac{\frac{2\sqrt{137}}{13}-\frac{304}{169}}{-\frac{98}{169}}$$

Divide -\frac{304}{169}+\frac{2\sqrt{137}}{13} by -\frac{98}{169} by multiplying -\frac{304}{169}+\frac{2\sqrt{137}}{13} by the reciprocal of -\frac{98}{169}.

$$x=\frac{152-13\sqrt{137}}{49}$$

Now solve the equation x=\frac{-\frac{304}{169}±\frac{2\sqrt{137}}{13}}{-\frac{98}{169}} when ± is minus. Subtract \frac{2\sqrt{137}}{13} from -\frac{304}{169}.

$$x=\frac{-\frac{2\sqrt{137}}{13}-\frac{304}{169}}{-\frac{98}{169}}$$

Divide -\frac{304}{169}-\frac{2\sqrt{137}}{13} by -\frac{98}{169} by multiplying -\frac{304}{169}-\frac{2\sqrt{137}}{13} by the reciprocal of -\frac{98}{169}.

$$x=\frac{13\sqrt{137}+152}{49}$$

The equation is now solved.

$$x=\frac{152-13\sqrt{137}}{49} x=\frac{13\sqrt{137}+152}{49}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(17x-1\right)^{2}.

$$\frac{289x^{2}-34x+1}{13^{2}}=\left(x-1\right)\times 2x$$

Calculate 13 to the power of 2 and get 169.

$$\frac{289x^{2}-34x+1}{169}=\left(x-1\right)\times 2x$$

Use the distributive property to multiply x-1 by 2.

$$\frac{289x^{2}-34x+1}{169}=\left(2x-2\right)x$$

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

$$\frac{289x^{2}-34x+1}{169}=2x^{2}-2x$$

Divide each term of 289x^{2}-34x+1 by 169 to get \frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=2x^{2}-2x$$

Subtract 2x^{2} from both sides.

$$\frac{289}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}-2x^{2}=-2x$$

Combine \frac{289}{169}x^{2} and -2x^{2} to get -\frac{49}{169}x^{2}.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}=-2x$$

Add 2x to both sides.

$$-\frac{49}{169}x^{2}-\frac{34}{169}x+\frac{1}{169}+2x=0$$

Combine -\frac{34}{169}x and 2x to get \frac{304}{169}x.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x+\frac{1}{169}=0$$

Subtract \frac{1}{169} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{49}{169}x^{2}+\frac{304}{169}x=-\frac{1}{169}$$

Divide both sides of the equation by -\frac{49}{169}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{49}{169}x^{2}+\frac{304}{169}x}{-\frac{49}{169}}=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Dividing by -\frac{49}{169} undoes the multiplication by -\frac{49}{169}.

$$x^{2}+\frac{\frac{304}{169}}{-\frac{49}{169}}x=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Divide \frac{304}{169} by -\frac{49}{169} by multiplying \frac{304}{169} by the reciprocal of -\frac{49}{169}.

$$x^{2}-\frac{304}{49}x=-\frac{\frac{1}{169}}{-\frac{49}{169}}$$

Divide -\frac{1}{169} by -\frac{49}{169} by multiplying -\frac{1}{169} by the reciprocal of -\frac{49}{169}.

$$x^{2}-\frac{304}{49}x=\frac{1}{49}$$

Divide -\frac{304}{49}, the coefficient of the x term, by 2 to get -\frac{152}{49}. Then add the square of -\frac{152}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{304}{49}x+\left(-\frac{152}{49}\right)^{2}=\frac{1}{49}+\left(-\frac{152}{49}\right)^{2}$$

Square -\frac{152}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{304}{49}x+\frac{23104}{2401}=\frac{1}{49}+\frac{23104}{2401}$$

Add \frac{1}{49} to \frac{23104}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{304}{49}x+\frac{23104}{2401}=\frac{23153}{2401}$$

Factor x^{2}-\frac{304}{49}x+\frac{23104}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{152}{49}\right)^{2}=\frac{23153}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{152}{49}\right)^{2}}=\sqrt{\frac{23153}{2401}}$$

Simplify.

$$x-\frac{152}{49}=\frac{13\sqrt{137}}{49} x-\frac{152}{49}=-\frac{13\sqrt{137}}{49}$$

Add \frac{152}{49} to both sides of the equation.

$$x=\frac{13\sqrt{137}+152}{49} x=\frac{152-13\sqrt{137}}{49}$$