Solve 35x^2-x-138=0

$$35 { x }^{ 2 } -x-138=0$$

Solve for x

$$x = -\frac{69}{35} = -1\frac{34}{35} \approx -1.971428571$$

$$x=2$$

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 35x^{2}+ax+bx-138. To find a and b, set up a system to be solved.

$$a+b=-1 ab=35\left(-138\right)=-4830$$

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 -4830.

$$1,-4830 2,-2415 3,-1610 5,-966 6,-805 7,-690 10,-483 14,-345 15,-322 21,-230 23,-210 30,-161 35,-138 42,-115 46,-105 69,-70$$

Calculate the sum for each pair.

$$1-4830=-4829 2-2415=-2413 3-1610=-1607 5-966=-961 6-805=-799 7-690=-683 10-483=-473 14-345=-331 15-322=-307 21-230=-209 23-210=-187 30-161=-131 35-138=-103 42-115=-73 46-105=-59 69-70=-1$$

The solution is the pair that gives sum -1.

$$a=-70 b=69$$

Rewrite 35x^{2}-x-138 as \left(35x^{2}-70x\right)+\left(69x-138\right).

$$\left(35x^{2}-70x\right)+\left(69x-138\right)$$

Factor out 35x in the first and 69 in the second group.

$$35x\left(x-2\right)+69\left(x-2\right)$$

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

$$\left(x-2\right)\left(35x+69\right)$$

To find equation solutions, solve x-2=0 and 35x+69=0.

$$x=2 x=-\frac{69}{35}$$

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.

$$35x^{2}-x-138=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, -1 for b, and -138 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 35\left(-138\right)}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-\left(-1\right)±\sqrt{1-140\left(-138\right)}}{2\times 35}$$

Multiply -140 times -138.

$$x=\frac{-\left(-1\right)±\sqrt{1+19320}}{2\times 35}$$

Add 1 to 19320.

$$x=\frac{-\left(-1\right)±\sqrt{19321}}{2\times 35}$$

Take the square root of 19321.

$$x=\frac{-\left(-1\right)±139}{2\times 35}$$

The opposite of -1 is 1.

$$x=\frac{1±139}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{1±139}{70}$$

Now solve the equation x=\frac{1±139}{70} when ± is plus. Add 1 to 139.

$$x=\frac{140}{70}$$

Divide 140 by 70.

$$x=2$$

Now solve the equation x=\frac{1±139}{70} when ± is minus. Subtract 139 from 1.

$$x=-\frac{138}{70}$$

Reduce the fraction \frac{-138}{70} to lowest terms by extracting and canceling out 2.

$$x=-\frac{69}{35}$$

The equation is now solved.

$$x=2 x=-\frac{69}{35}$$

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.

$$35x^{2}-x-138=0$$

Add 138 to both sides of the equation.

$$35x^{2}-x-138-\left(-138\right)=-\left(-138\right)$$

Subtracting -138 from itself leaves 0.

$$35x^{2}-x=-\left(-138\right)$$

Subtract -138 from 0.

$$35x^{2}-x=138$$

Divide both sides by 35.

$$\frac{35x^{2}-x}{35}=\frac{138}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}-\frac{1}{35}x=\frac{138}{35}$$

Divide -\frac{1}{35}, the coefficient of the x term, by 2 to get -\frac{1}{70}. Then add the square of -\frac{1}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{35}x+\left(-\frac{1}{70}\right)^{2}=\frac{138}{35}+\left(-\frac{1}{70}\right)^{2}$$

Square -\frac{1}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{35}x+\frac{1}{4900}=\frac{138}{35}+\frac{1}{4900}$$

Add \frac{138}{35} to \frac{1}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{35}x+\frac{1}{4900}=\frac{19321}{4900}$$

Factor x^{2}-\frac{1}{35}x+\frac{1}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{70}\right)^{2}=\frac{19321}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{70}\right)^{2}}=\sqrt{\frac{19321}{4900}}$$

Simplify.

$$x-\frac{1}{70}=\frac{139}{70} x-\frac{1}{70}=-\frac{139}{70}$$

Add \frac{1}{70} to both sides of the equation.

$$x=2 x=-\frac{69}{35}$$