Solve 17x^2-6x-11=0

$$17 { x }^{ 2 } -6x-11 = 0$$

Solve for x

$$x=-\frac{11}{17}\approx -0.647058824$$

$$x=1$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 17x^{2}+ax+bx-11. To find a and b, set up a system to be solved.

$$a+b=-6 ab=17\left(-11\right)=-187$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -187.

$$1,-187 11,-17$$

Calculate the sum for each pair.

$$1-187=-186 11-17=-6$$

The solution is the pair that gives sum -6.

$$a=-17 b=11$$

Rewrite 17x^{2}-6x-11 as \left(17x^{2}-17x\right)+\left(11x-11\right).

$$\left(17x^{2}-17x\right)+\left(11x-11\right)$$

Factor out 17x in the first and 11 in the second group.

$$17x\left(x-1\right)+11\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(17x+11\right)$$

To find equation solutions, solve x-1=0 and 17x+11=0.

$$x=1 x=-\frac{11}{17}$$

Steps Using the Quadratic Formula

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.

$$17x^{2}-6x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 17 for a, -6 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 17\left(-11\right)}}{2\times 17}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 17\left(-11\right)}}{2\times 17}$$

Multiply -4 times 17.

$$x=\frac{-\left(-6\right)±\sqrt{36-68\left(-11\right)}}{2\times 17}$$

Multiply -68 times -11.

$$x=\frac{-\left(-6\right)±\sqrt{36+748}}{2\times 17}$$

Add 36 to 748.

$$x=\frac{-\left(-6\right)±\sqrt{784}}{2\times 17}$$

Take the square root of 784.

$$x=\frac{-\left(-6\right)±28}{2\times 17}$$

The opposite of -6 is 6.

$$x=\frac{6±28}{2\times 17}$$

Multiply 2 times 17.

$$x=\frac{6±28}{34}$$

Now solve the equation x=\frac{6±28}{34} when ± is plus. Add 6 to 28.

$$x=\frac{34}{34}$$

Divide 34 by 34.

$$x=1$$

Now solve the equation x=\frac{6±28}{34} when ± is minus. Subtract 28 from 6.

$$x=-\frac{22}{34}$$

Reduce the fraction \frac{-22}{34} to lowest terms by extracting and canceling out 2.

$$x=-\frac{11}{17}$$

The equation is now solved.

$$x=1 x=-\frac{11}{17}$$

Steps for Completing the Square

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.

$$17x^{2}-6x-11=0$$

Add 11 to both sides of the equation.

$$17x^{2}-6x-11-\left(-11\right)=-\left(-11\right)$$

Subtracting -11 from itself leaves 0.

$$17x^{2}-6x=-\left(-11\right)$$

Subtract -11 from 0.

$$17x^{2}-6x=11$$

Divide both sides by 17.

$$\frac{17x^{2}-6x}{17}=\frac{11}{17}$$

Dividing by 17 undoes the multiplication by 17.

$$x^{2}-\frac{6}{17}x=\frac{11}{17}$$

Divide -\frac{6}{17}, the coefficient of the x term, by 2 to get -\frac{3}{17}. Then add the square of -\frac{3}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{17}x+\left(-\frac{3}{17}\right)^{2}=\frac{11}{17}+\left(-\frac{3}{17}\right)^{2}$$

Square -\frac{3}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{11}{17}+\frac{9}{289}$$

Add \frac{11}{17} to \frac{9}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{196}{289}$$

Factor x^{2}-\frac{6}{17}x+\frac{9}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{17}\right)^{2}=\frac{196}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{17}\right)^{2}}=\sqrt{\frac{196}{289}}$$

Simplify.

$$x-\frac{3}{17}=\frac{14}{17} x-\frac{3}{17}=-\frac{14}{17}$$

Add \frac{3}{17} to both sides of the equation.

$$x=1 x=-\frac{11}{17}$$