Solve 169x^2-130x-231=0

$$169 x ^ { 2 } - 130 x - 231 = 0$$

Solve for x

$$x=-\frac{11}{13}\approx -0.846153846$$

$$x = \frac{21}{13} = 1\frac{8}{13} \approx 1.615384615$$

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 169x^{2}+ax+bx-231. To find a and b, set up a system to be solved.

$$a+b=-130 ab=169\left(-231\right)=-39039$$

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 -39039.

$$1,-39039 3,-13013 7,-5577 11,-3549 13,-3003 21,-1859 33,-1183 39,-1001 77,-507 91,-429 143,-273 169,-231$$

Calculate the sum for each pair.

$$1-39039=-39038 3-13013=-13010 7-5577=-5570 11-3549=-3538 13-3003=-2990 21-1859=-1838 33-1183=-1150 39-1001=-962 77-507=-430 91-429=-338 143-273=-130 169-231=-62$$

The solution is the pair that gives sum -130.

$$a=-273 b=143$$

Rewrite 169x^{2}-130x-231 as \left(169x^{2}-273x\right)+\left(143x-231\right).

$$\left(169x^{2}-273x\right)+\left(143x-231\right)$$

Factor out 13x in the first and 11 in the second group.

$$13x\left(13x-21\right)+11\left(13x-21\right)$$

Factor out common term 13x-21 by using distributive property.

$$\left(13x-21\right)\left(13x+11\right)$$

To find equation solutions, solve 13x-21=0 and 13x+11=0.

$$x=\frac{21}{13} x=-\frac{11}{13}$$

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.

$$169x^{2}-130x-231=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 169 for a, -130 for b, and -231 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-130\right)±\sqrt{\left(-130\right)^{2}-4\times 169\left(-231\right)}}{2\times 169}$$

Square -130.

$$x=\frac{-\left(-130\right)±\sqrt{16900-4\times 169\left(-231\right)}}{2\times 169}$$

Multiply -4 times 169.

$$x=\frac{-\left(-130\right)±\sqrt{16900-676\left(-231\right)}}{2\times 169}$$

Multiply -676 times -231.

$$x=\frac{-\left(-130\right)±\sqrt{16900+156156}}{2\times 169}$$

Add 16900 to 156156.

$$x=\frac{-\left(-130\right)±\sqrt{173056}}{2\times 169}$$

Take the square root of 173056.

$$x=\frac{-\left(-130\right)±416}{2\times 169}$$

The opposite of -130 is 130.

$$x=\frac{130±416}{2\times 169}$$

Multiply 2 times 169.

$$x=\frac{130±416}{338}$$

Now solve the equation x=\frac{130±416}{338} when ± is plus. Add 130 to 416.

$$x=\frac{546}{338}$$

Reduce the fraction \frac{546}{338} to lowest terms by extracting and canceling out 26.

$$x=\frac{21}{13}$$

Now solve the equation x=\frac{130±416}{338} when ± is minus. Subtract 416 from 130.

$$x=-\frac{286}{338}$$

Reduce the fraction \frac{-286}{338} to lowest terms by extracting and canceling out 26.

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

The equation is now solved.

$$x=\frac{21}{13} x=-\frac{11}{13}$$

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.

$$169x^{2}-130x-231=0$$

Add 231 to both sides of the equation.

$$169x^{2}-130x-231-\left(-231\right)=-\left(-231\right)$$

Subtracting -231 from itself leaves 0.

$$169x^{2}-130x=-\left(-231\right)$$

Subtract -231 from 0.

$$169x^{2}-130x=231$$

Divide both sides by 169.

$$\frac{169x^{2}-130x}{169}=\frac{231}{169}$$

Dividing by 169 undoes the multiplication by 169.

$$x^{2}+\left(-\frac{130}{169}\right)x=\frac{231}{169}$$

Reduce the fraction \frac{-130}{169} to lowest terms by extracting and canceling out 13.

$$x^{2}-\frac{10}{13}x=\frac{231}{169}$$

Divide -\frac{10}{13}, the coefficient of the x term, by 2 to get -\frac{5}{13}. Then add the square of -\frac{5}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{13}x+\left(-\frac{5}{13}\right)^{2}=\frac{231}{169}+\left(-\frac{5}{13}\right)^{2}$$

Square -\frac{5}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{13}x+\frac{25}{169}=\frac{231+25}{169}$$

Add \frac{231}{169} to \frac{25}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{10}{13}x+\frac{25}{169}=\frac{256}{169}$$

Factor x^{2}-\frac{10}{13}x+\frac{25}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{13}\right)^{2}=\frac{256}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{13}\right)^{2}}=\sqrt{\frac{256}{169}}$$

Simplify.

$$x-\frac{5}{13}=\frac{16}{13} x-\frac{5}{13}=-\frac{16}{13}$$

Add \frac{5}{13} to both sides of the equation.

$$x=\frac{21}{13} x=-\frac{11}{13}$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 169

$$x ^ 2 -\frac{10}{13}x -\frac{231}{169} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{10}{13} rs = -\frac{231}{169}$$

Two numbers r and s sum up to \frac{10}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{10}{13} = \frac{5}{13}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{5}{13} - u s = \frac{5}{13} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{231}{169}

$$(\frac{5}{13} - u) (\frac{5}{13} + u) = -\frac{231}{169}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{25}{169} - u^2 = -\frac{231}{169}$$

Simplify the expression by subtracting \frac{25}{169} on both sides

$$-u^2 = -\frac{231}{169}-\frac{25}{169} = -\frac{256}{169}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{256}{169} u = \pm\sqrt{\frac{256}{169}} = \pm \frac{16}{13}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{5}{13} - \frac{16}{13} = -0.846 s = \frac{5}{13} + \frac{16}{13} = 1.615$$