Solve 12x^2-42x+30

$$12 { x }^{ 2 } -42x+30$$

Factor

$$6\left(x-1\right)\left(2x-5\right)$$

Solution Steps

Factor out 6.

$$6\left(2x^{2}-7x+5\right)$$

Consider 2x^{2}-7x+5. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\times 5=10$$

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 10.

$$-1,-10 -2,-5$$

Calculate the sum for each pair.

$$-1-10=-11 -2-5=-7$$

The solution is the pair that gives sum -7.

$$a=-5 b=-2$$

Rewrite 2x^{2}-7x+5 as \left(2x^{2}-5x\right)+\left(-2x+5\right).

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

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

$$x\left(2x-5\right)-\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$6\left(2x-5\right)\left(x-1\right)$$

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.

$$12x^{2}-42x+30=0$$

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.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 12\times 30}}{2\times 12}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\times 12\times 30}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-42\right)±\sqrt{1764-48\times 30}}{2\times 12}$$

Multiply -48 times 30.

$$x=\frac{-\left(-42\right)±\sqrt{1764-1440}}{2\times 12}$$

Add 1764 to -1440.

$$x=\frac{-\left(-42\right)±\sqrt{324}}{2\times 12}$$

Take the square root of 324.

$$x=\frac{-\left(-42\right)±18}{2\times 12}$$

The opposite of -42 is 42.

$$x=\frac{42±18}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{42±18}{24}$$

Now solve the equation x=\frac{42±18}{24} when ± is plus. Add 42 to 18.

$$x=\frac{60}{24}$$

Reduce the fraction \frac{60}{24} to lowest terms by extracting and canceling out 12.

$$x=\frac{5}{2}$$

Now solve the equation x=\frac{42±18}{24} when ± is minus. Subtract 18 from 42.

$$x=\frac{24}{24}$$

Divide 24 by 24.

$$x=1$$

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

$$12x^{2}-42x+30=12\left(x-\frac{5}{2}\right)\left(x-1\right)$$

Subtract \frac{5}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$12x^{2}-42x+30=12\times \frac{2x-5}{2}\left(x-1\right)$$

Cancel out 2, the greatest common factor in 12 and 2.

$$12x^{2}-42x+30=6\left(2x-5\right)\left(x-1\right)$$

Evaluate

$$6\left(x-1\right)\left(2x-5\right)$$