Solve 72x^2-12x-24=0

$$72 { x }^{ 2 } -12x-24=0$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Divide both sides by 12.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-1 ab=6\left(-2\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=-4 b=3$$

Rewrite 6x^{2}-x-2 as \left(6x^{2}-4x\right)+\left(3x-2\right).

$$\left(6x^{2}-4x\right)+\left(3x-2\right)$$

Factor out 2x in 6x^{2}-4x.

$$2x\left(3x-2\right)+3x-2$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(2x+1\right)$$

To find equation solutions, solve 3x-2=0 and 2x+1=0.

$$x=\frac{2}{3} x=-\frac{1}{2}$$

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.

$$72x^{2}-12x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 72 for a, -12 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 72\left(-24\right)}}{2\times 72}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 72\left(-24\right)}}{2\times 72}$$

Multiply -4 times 72.

$$x=\frac{-\left(-12\right)±\sqrt{144-288\left(-24\right)}}{2\times 72}$$

Multiply -288 times -24.

$$x=\frac{-\left(-12\right)±\sqrt{144+6912}}{2\times 72}$$

Add 144 to 6912.

$$x=\frac{-\left(-12\right)±\sqrt{7056}}{2\times 72}$$

Take the square root of 7056.

$$x=\frac{-\left(-12\right)±84}{2\times 72}$$

The opposite of -12 is 12.

$$x=\frac{12±84}{2\times 72}$$

Multiply 2 times 72.

$$x=\frac{12±84}{144}$$

Now solve the equation x=\frac{12±84}{144} when ± is plus. Add 12 to 84.

$$x=\frac{96}{144}$$

Reduce the fraction \frac{96}{144} to lowest terms by extracting and canceling out 48.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{12±84}{144} when ± is minus. Subtract 84 from 12.

$$x=-\frac{72}{144}$$

Reduce the fraction \frac{-72}{144} to lowest terms by extracting and canceling out 72.

$$x=-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{2}{3} x=-\frac{1}{2}$$

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.

$$72x^{2}-12x-24=0$$

Add 24 to both sides of the equation.

$$72x^{2}-12x-24-\left(-24\right)=-\left(-24\right)$$

Subtracting -24 from itself leaves 0.

$$72x^{2}-12x=-\left(-24\right)$$

Subtract -24 from 0.

$$72x^{2}-12x=24$$

Divide both sides by 72.

$$\frac{72x^{2}-12x}{72}=\frac{24}{72}$$

Dividing by 72 undoes the multiplication by 72.

$$x^{2}+\left(-\frac{12}{72}\right)x=\frac{24}{72}$$

Reduce the fraction \frac{-12}{72} to lowest terms by extracting and canceling out 12.

$$x^{2}-\frac{1}{6}x=\frac{24}{72}$$

Reduce the fraction \frac{24}{72} to lowest terms by extracting and canceling out 24.

$$x^{2}-\frac{1}{6}x=\frac{1}{3}$$

Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=\frac{1}{3}+\left(-\frac{1}{12}\right)^{2}$$

Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{1}{3}+\frac{1}{144}$$

Add \frac{1}{3} to \frac{1}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{49}{144}$$

Factor x^{2}-\frac{1}{6}x+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{1}{12}=\frac{7}{12} x-\frac{1}{12}=-\frac{7}{12}$$

Add \frac{1}{12} to both sides of the equation.

$$x=\frac{2}{3} x=-\frac{1}{2}$$