Solve -1+6d^2-5d

$$- 1 + 6 d ^ { 2 } - 5 d$$

Factor

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

Solution Steps

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

$$6d^{2}-5d-1$$

Factor the expression by grouping. First, the expression needs to be rewritten as 6d^{2}+ad+bd-1. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-1\right)=-6$$

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 -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

Rewrite 6d^{2}-5d-1 as \left(6d^{2}-6d\right)+\left(d-1\right).

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

Factor out 6d in 6d^{2}-6d.

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

Factor out common term d-1 by using distributive property.

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

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.

$$6d^{2}-5d-1=0$$

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.

$$d=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -5.

$$d=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$d=\frac{-\left(-5\right)±\sqrt{25-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$d=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 6}$$

Add 25 to 24.

$$d=\frac{-\left(-5\right)±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$d=\frac{-\left(-5\right)±7}{2\times 6}$$

The opposite of -5 is 5.

$$d=\frac{5±7}{2\times 6}$$

Multiply 2 times 6.

$$d=\frac{5±7}{12}$$

Now solve the equation d=\frac{5±7}{12} when ± is plus. Add 5 to 7.

$$d=\frac{12}{12}$$

Divide 12 by 12.

$$d=1$$

Now solve the equation d=\frac{5±7}{12} when ± is minus. Subtract 7 from 5.

$$d=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$d=-\frac{1}{6}$$

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

$$6d^{2}-5d-1=6\left(d-1\right)\left(d-\left(-\frac{1}{6}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$6d^{2}-5d-1=6\left(d-1\right)\left(d+\frac{1}{6}\right)$$

Add \frac{1}{6} to d by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6d^{2}-5d-1=6\left(d-1\right)\times \frac{6d+1}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6d^{2}-5d-1=\left(d-1\right)\left(6d+1\right)$$

Evaluate

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