Solve 6x^2-30x+24=

$$6 { x }^{ 2 } -30x+24=$$

Factor

$$6\left(x-4\right)\left(x-1\right)$$

Solution Steps

Factor out 6.

$$6\left(x^{2}-5x+4\right)$$

Consider x^{2}-5x+4. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 4=4$$

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 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite x^{2}-5x+4 as \left(x^{2}-4x\right)+\left(-x+4\right).

$$\left(x^{2}-4x\right)+\left(-x+4\right)$$

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

$$x\left(x-4\right)-\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$6\left(x-4\right)\left(x-1\right)$$

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.

$$6x^{2}-30x+24=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 6\times 24}}{2\times 6}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 6\times 24}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-30\right)±\sqrt{900-24\times 24}}{2\times 6}$$

Multiply -24 times 24.

$$x=\frac{-\left(-30\right)±\sqrt{900-576}}{2\times 6}$$

Add 900 to -576.

$$x=\frac{-\left(-30\right)±\sqrt{324}}{2\times 6}$$

Take the square root of 324.

$$x=\frac{-\left(-30\right)±18}{2\times 6}$$

The opposite of -30 is 30.

$$x=\frac{30±18}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{30±18}{12}$$

Now solve the equation x=\frac{30±18}{12} when ± is plus. Add 30 to 18.

$$x=\frac{48}{12}$$

Divide 48 by 12.

$$x=4$$

Now solve the equation x=\frac{30±18}{12} when ± is minus. Subtract 18 from 30.

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

Divide 12 by 12.

$$x=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and 1 for x_{2}.

$$6x^{2}-30x+24=6\left(x-4\right)\left(x-1\right)$$

Evaluate

$$6\left(x-4\right)\left(x-1\right)$$