Solve 5w^2-6w+1

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

Factor

$$\left(w-1\right)\left(5w-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5w^{2}+aw+bw+1. To find a and b, set up a system to be solved.

$$a+b=-6 ab=5\times 1=5$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-5 b=-1$$

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

$$\left(5w^{2}-5w\right)+\left(-w+1\right)$$

Factor out 5w in the first and -1 in the second group.

$$5w\left(w-1\right)-\left(w-1\right)$$

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

$$\left(w-1\right)\left(5w-1\right)$$

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.

$$5w^{2}-6w+1=0$$

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.

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

Square -6.

$$w=\frac{-\left(-6\right)±\sqrt{36-4\times 5}}{2\times 5}$$

Multiply -4 times 5.

$$w=\frac{-\left(-6\right)±\sqrt{36-20}}{2\times 5}$$

Add 36 to -20.

$$w=\frac{-\left(-6\right)±\sqrt{16}}{2\times 5}$$

Take the square root of 16.

$$w=\frac{-\left(-6\right)±4}{2\times 5}$$

The opposite of -6 is 6.

$$w=\frac{6±4}{2\times 5}$$

Multiply 2 times 5.

$$w=\frac{6±4}{10}$$

Now solve the equation w=\frac{6±4}{10} when ± is plus. Add 6 to 4.

$$w=\frac{10}{10}$$

Divide 10 by 10.

$$w=1$$

Now solve the equation w=\frac{6±4}{10} when ± is minus. Subtract 4 from 6.

$$w=\frac{2}{10}$$

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

$$w=\frac{1}{5}$$

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}{5} for x_{2}.

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

Subtract \frac{1}{5} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Cancel out 5, the greatest common factor in 5 and 5.

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

Evaluate

$$\left(w-1\right)\left(5w-1\right)$$