Solve (t-3t^2)*12-1=0

$$( t - 3 t ^ { 2 } ) \cdot 12 - 1 = 0$$

Solve for t

$$t=\frac{1}{6}\approx 0.166666667$$

Steps Using Factoring By Grouping

Use the distributive property to multiply t-3t^{2} by 12.

$$12t-36t^{2}-1=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-36t^{2}+12t-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -36t^{2}+at+bt-1. To find a and b, set up a system to be solved.

$$a+b=12 ab=-36\left(-1\right)=36$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 12.

$$a=6 b=6$$

Rewrite -36t^{2}+12t-1 as \left(-36t^{2}+6t\right)+\left(6t-1\right).

$$\left(-36t^{2}+6t\right)+\left(6t-1\right)$$

Factor out -6t in -36t^{2}+6t.

$$-6t\left(6t-1\right)+6t-1$$

Factor out common term 6t-1 by using distributive property.

$$\left(6t-1\right)\left(-6t+1\right)$$

To find equation solutions, solve 6t-1=0 and -6t+1=0.

$$t=\frac{1}{6} t=\frac{1}{6}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply t-3t^{2} by 12.

$$12t-36t^{2}-1=0$$

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.

$$-36t^{2}+12t-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, 12 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-12±\sqrt{12^{2}-4\left(-36\right)\left(-1\right)}}{2\left(-36\right)}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\left(-36\right)\left(-1\right)}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$t=\frac{-12±\sqrt{144+144\left(-1\right)}}{2\left(-36\right)}$$

Multiply 144 times -1.

$$t=\frac{-12±\sqrt{144-144}}{2\left(-36\right)}$$

Add 144 to -144.

$$t=\frac{-12±\sqrt{0}}{2\left(-36\right)}$$

Take the square root of 0.

$$t=-\frac{12}{2\left(-36\right)}$$

Multiply 2 times -36.

$$t=-\frac{12}{-72}$$

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

$$t=\frac{1}{6}$$

Steps for Completing the Square

Use the distributive property to multiply t-3t^{2} by 12.

$$12t-36t^{2}-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$12t-36t^{2}=1$$

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.

$$-36t^{2}+12t=1$$

Divide both sides by -36.

$$\frac{-36t^{2}+12t}{-36}=\frac{1}{-36}$$

Dividing by -36 undoes the multiplication by -36.

$$t^{2}+\frac{12}{-36}t=\frac{1}{-36}$$

Reduce the fraction \frac{12}{-36} to lowest terms by extracting and canceling out 12.

$$t^{2}-\frac{1}{3}t=\frac{1}{-36}$$

Divide 1 by -36.

$$t^{2}-\frac{1}{3}t=-\frac{1}{36}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1}{3}t+\left(-\frac{1}{6}\right)^{2}=-\frac{1}{36}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1}{3}t+\frac{1}{36}=\frac{-1+1}{36}$$

Add -\frac{1}{36} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1}{3}t+\frac{1}{36}=0$$

Factor t^{2}-\frac{1}{3}t+\frac{1}{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}.

$$\left(t-\frac{1}{6}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{1}{6}\right)^{2}}=\sqrt{0}$$

Simplify.

$$t-\frac{1}{6}=0 t-\frac{1}{6}=0$$

Add \frac{1}{6} to both sides of the equation.

$$t=\frac{1}{6} t=\frac{1}{6}$$

The equation is now solved. Solutions are the same.

$$t=\frac{1}{6}$$