Solve 8x+30x^2-6

$$8 x + 30 x ^ { 2 } - 6$$

Factor

$$2\left(3x-1\right)\left(5x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(4x+15x^{2}-3\right)$$

Consider 4x+15x^{2}-3. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$15x^{2}+4x-3$$

Factor the expression by grouping. First, the expression needs to be rewritten as 15x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=15\left(-3\right)=-45$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -45.

$$-1,45 -3,15 -5,9$$

Calculate the sum for each pair.

$$-1+45=44 -3+15=12 -5+9=4$$

The solution is the pair that gives sum 4.

$$a=-5 b=9$$

Rewrite 15x^{2}+4x-3 as \left(15x^{2}-5x\right)+\left(9x-3\right).

$$\left(15x^{2}-5x\right)+\left(9x-3\right)$$

Factor out 5x in the first and 3 in the second group.

$$5x\left(3x-1\right)+3\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(5x+3\right)$$

Rewrite the complete factored expression.

$$2\left(3x-1\right)\left(5x+3\right)$$

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.

$$30x^{2}+8x-6=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\times 30\left(-6\right)}}{2\times 30}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 30\left(-6\right)}}{2\times 30}$$

Multiply -4 times 30.

$$x=\frac{-8±\sqrt{64-120\left(-6\right)}}{2\times 30}$$

Multiply -120 times -6.

$$x=\frac{-8±\sqrt{64+720}}{2\times 30}$$

Add 64 to 720.

$$x=\frac{-8±\sqrt{784}}{2\times 30}$$

Take the square root of 784.

$$x=\frac{-8±28}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{-8±28}{60}$$

Now solve the equation x=\frac{-8±28}{60} when ± is plus. Add -8 to 28.

$$x=\frac{20}{60}$$

Reduce the fraction \frac{20}{60} to lowest terms by extracting and canceling out 20.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-8±28}{60} when ± is minus. Subtract 28 from -8.

$$x=-\frac{36}{60}$$

Reduce the fraction \frac{-36}{60} to lowest terms by extracting and canceling out 12.

$$x=-\frac{3}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{3} for x_{1} and -\frac{3}{5} for x_{2}.

$$30x^{2}+8x-6=30\left(x-\frac{1}{3}\right)\left(x-\left(-\frac{3}{5}\right)\right)$$

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

$$30x^{2}+8x-6=30\left(x-\frac{1}{3}\right)\left(x+\frac{3}{5}\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$30x^{2}+8x-6=30\times \frac{3x-1}{3}\left(x+\frac{3}{5}\right)$$

Add \frac{3}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$30x^{2}+8x-6=30\times \frac{3x-1}{3}\times \frac{5x+3}{5}$$

Multiply \frac{3x-1}{3} times \frac{5x+3}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$30x^{2}+8x-6=30\times \frac{\left(3x-1\right)\left(5x+3\right)}{3\times 5}$$

Multiply 3 times 5.

$$30x^{2}+8x-6=30\times \frac{\left(3x-1\right)\left(5x+3\right)}{15}$$

Cancel out 15, the greatest common factor in 30 and 15.

$$30x^{2}+8x-6=2\left(3x-1\right)\left(5x+3\right)$$

Evaluate

$$30x^{2}+8x-6$$