Solve -36x^2+12x-1

$$-36 { x }^{ 2 } +12x-1$$

Factor

$$-\left(6x-1\right)^{2}$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -36x^{2}+ax+bx-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 -36x^{2}+12x-1 as \left(-36x^{2}+6x\right)+\left(6x-1\right).

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

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

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

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

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

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

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

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.

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

Square 12.

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

Multiply -4 times -36.

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

Multiply 144 times -1.

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

Add 144 to -144.

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

Take the square root of 0.

$$x=\frac{-12±0}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{-12±0}{-72}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{6} for x_{1} and \frac{1}{6} for x_{2}.

$$-36x^{2}+12x-1=-36\left(x-\frac{1}{6}\right)\left(x-\frac{1}{6}\right)$$

Subtract \frac{1}{6} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-36x^{2}+12x-1=-36\times \frac{-6x+1}{-6}\left(x-\frac{1}{6}\right)$$

Subtract \frac{1}{6} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-36x^{2}+12x-1=-36\times \frac{-6x+1}{-6}\times \frac{-6x+1}{-6}$$

Multiply \frac{-6x+1}{-6} times \frac{-6x+1}{-6} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-36x^{2}+12x-1=-36\times \frac{\left(-6x+1\right)\left(-6x+1\right)}{-6\left(-6\right)}$$

Multiply -6 times -6.

$$-36x^{2}+12x-1=-36\times \frac{\left(-6x+1\right)\left(-6x+1\right)}{36}$$

Cancel out 36, the greatest common factor in -36 and 36.

$$-36x^{2}+12x-1=-\left(-6x+1\right)\left(-6x+1\right)$$

Evaluate

$$-\left(6x-1\right)^{2}$$