Solve 30x^2-246x-216

$$30 { x }^{ 2 } -246x-216$$

Factor

$$6\left(x-9\right)\left(5x+4\right)$$

Solution Steps

Factor out 6.

$$6\left(5x^{2}-41x-36\right)$$

Consider 5x^{2}-41x-36. Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-41 ab=5\left(-36\right)=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Calculate the sum for each pair.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

The solution is the pair that gives sum -41.

$$a=-45 b=4$$

Rewrite 5x^{2}-41x-36 as \left(5x^{2}-45x\right)+\left(4x-36\right).

$$\left(5x^{2}-45x\right)+\left(4x-36\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(x-9\right)+4\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(5x+4\right)$$

Rewrite the complete factored expression.

$$6\left(x-9\right)\left(5x+4\right)$$

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.

$$30x^{2}-246x-216=0$$

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.

$$x=\frac{-\left(-246\right)±\sqrt{\left(-246\right)^{2}-4\times 30\left(-216\right)}}{2\times 30}$$

Square -246.

$$x=\frac{-\left(-246\right)±\sqrt{60516-4\times 30\left(-216\right)}}{2\times 30}$$

Multiply -4 times 30.

$$x=\frac{-\left(-246\right)±\sqrt{60516-120\left(-216\right)}}{2\times 30}$$

Multiply -120 times -216.

$$x=\frac{-\left(-246\right)±\sqrt{60516+25920}}{2\times 30}$$

Add 60516 to 25920.

$$x=\frac{-\left(-246\right)±\sqrt{86436}}{2\times 30}$$

Take the square root of 86436.

$$x=\frac{-\left(-246\right)±294}{2\times 30}$$

The opposite of -246 is 246.

$$x=\frac{246±294}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{246±294}{60}$$

Now solve the equation x=\frac{246±294}{60} when ± is plus. Add 246 to 294.

$$x=\frac{540}{60}$$

Divide 540 by 60.

$$x=9$$

Now solve the equation x=\frac{246±294}{60} when ± is minus. Subtract 294 from 246.

$$x=-\frac{48}{60}$$

Reduce the fraction \frac{-48}{60} to lowest terms by extracting and canceling out 12.

$$x=-\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -\frac{4}{5} for x_{2}.

$$30x^{2}-246x-216=30\left(x-9\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

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

$$30x^{2}-246x-216=30\left(x-9\right)\left(x+\frac{4}{5}\right)$$

Add \frac{4}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$30x^{2}-246x-216=30\left(x-9\right)\times \frac{5x+4}{5}$$

Cancel out 5, the greatest common factor in 30 and 5.

$$30x^{2}-246x-216=6\left(x-9\right)\left(5x+4\right)$$

Evaluate

$$6\left(x-9\right)\left(5x+4\right)$$