Solve 3x^2-31x-60=0

$$3 { x }^{ 2 } -31x-60=0$$

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$x=12$$

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 3x^{2}+ax+bx-60. To find a and b, set up a system to be solved.

$$a+b=-31 ab=3\left(-60\right)=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Calculate the sum for each pair.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

The solution is the pair that gives sum -31.

$$a=-36 b=5$$

Rewrite 3x^{2}-31x-60 as \left(3x^{2}-36x\right)+\left(5x-60\right).

$$\left(3x^{2}-36x\right)+\left(5x-60\right)$$

Factor out 3x in the first and 5 in the second group.

$$3x\left(x-12\right)+5\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(3x+5\right)$$

To find equation solutions, solve x-12=0 and 3x+5=0.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -31 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 3\left(-60\right)}}{2\times 3}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 3\left(-60\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-31\right)±\sqrt{961-12\left(-60\right)}}{2\times 3}$$

Multiply -12 times -60.

$$x=\frac{-\left(-31\right)±\sqrt{961+720}}{2\times 3}$$

Add 961 to 720.

$$x=\frac{-\left(-31\right)±\sqrt{1681}}{2\times 3}$$

Take the square root of 1681.

$$x=\frac{-\left(-31\right)±41}{2\times 3}$$

The opposite of -31 is 31.

$$x=\frac{31±41}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{31±41}{6}$$

Now solve the equation x=\frac{31±41}{6} when ± is plus. Add 31 to 41.

$$x=\frac{72}{6}$$

Divide 72 by 6.

$$x=12$$

Now solve the equation x=\frac{31±41}{6} when ± is minus. Subtract 41 from 31.

$$x=-\frac{10}{6}$$

Reduce the fraction \frac{-10}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{3}$$

The equation is now solved.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

Add 60 to both sides of the equation.

$$3x^{2}-31x-60-\left(-60\right)=-\left(-60\right)$$

Subtracting -60 from itself leaves 0.

$$3x^{2}-31x=-\left(-60\right)$$

Subtract -60 from 0.

$$3x^{2}-31x=60$$

Divide both sides by 3.

$$\frac{3x^{2}-31x}{3}=\frac{60}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{31}{3}x=\frac{60}{3}$$

Divide 60 by 3.

$$x^{2}-\frac{31}{3}x=20$$

Divide -\frac{31}{3}, the coefficient of the x term, by 2 to get -\frac{31}{6}. Then add the square of -\frac{31}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{3}x+\left(-\frac{31}{6}\right)^{2}=20+\left(-\frac{31}{6}\right)^{2}$$

Square -\frac{31}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=20+\frac{961}{36}$$

Add 20 to \frac{961}{36}.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=\frac{1681}{36}$$

Factor x^{2}-\frac{31}{3}x+\frac{961}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{6}\right)^{2}=\frac{1681}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{6}\right)^{2}}=\sqrt{\frac{1681}{36}}$$

Simplify.

$$x-\frac{31}{6}=\frac{41}{6} x-\frac{31}{6}=-\frac{41}{6}$$

Add \frac{31}{6} to both sides of the equation.

$$x=12 x=-\frac{5}{3}$$