Solve 5x^2-11x+6

$$5 { x }^{ 2 } -11x+6$$

Factor

$$\left(x-1\right)\left(5x-6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-11 ab=5\times 6=30$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -11.

$$a=-6 b=-5$$

Rewrite 5x^{2}-11x+6 as \left(5x^{2}-6x\right)+\left(-5x+6\right).

$$\left(5x^{2}-6x\right)+\left(-5x+6\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(5x-6\right)-\left(5x-6\right)$$

Factor out common term 5x-6 by using distributive property.

$$\left(5x-6\right)\left(x-1\right)$$

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.

$$5x^{2}-11x+6=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 5\times 6}}{2\times 5}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 5\times 6}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-11\right)±\sqrt{121-20\times 6}}{2\times 5}$$

Multiply -20 times 6.

$$x=\frac{-\left(-11\right)±\sqrt{121-120}}{2\times 5}$$

Add 121 to -120.

$$x=\frac{-\left(-11\right)±\sqrt{1}}{2\times 5}$$

Take the square root of 1.

$$x=\frac{-\left(-11\right)±1}{2\times 5}$$

The opposite of -11 is 11.

$$x=\frac{11±1}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{11±1}{10}$$

Now solve the equation x=\frac{11±1}{10} when ± is plus. Add 11 to 1.

$$x=\frac{12}{10}$$

Reduce the fraction \frac{12}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{6}{5}$$

Now solve the equation x=\frac{11±1}{10} when ± is minus. Subtract 1 from 11.

$$x=\frac{10}{10}$$

Divide 10 by 10.

$$x=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{6}{5} for x_{1} and 1 for x_{2}.

$$5x^{2}-11x+6=5\left(x-\frac{6}{5}\right)\left(x-1\right)$$

Subtract \frac{6}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}-11x+6=5\times \frac{5x-6}{5}\left(x-1\right)$$

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

$$5x^{2}-11x+6=\left(5x-6\right)\left(x-1\right)$$

Evaluate

$$\left(x-1\right)\left(5x-6\right)$$