Solve (6x^2-8x+5)+(2x-13)

$$( 6 x ^ { 2 } - 8 x + 5 ) + ( 2 x - 13 )$$

Evaluate

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

Solution Steps

Combine -8x and 2x to get -6x.

$$6x^{2}-6x+5-13$$

Subtract 13 from 5 to get -8.

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

Factor

$$6\left(x-\left(-\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine -8x and 2x to get -6x.

$$factor(6x^{2}-6x+5-13)$$

Subtract 13 from 5 to get -8.

$$factor(6x^{2}-6x-8)$$

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.

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

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.

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

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 6\left(-8\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36-24\left(-8\right)}}{2\times 6}$$

Multiply -24 times -8.

$$x=\frac{-\left(-6\right)±\sqrt{36+192}}{2\times 6}$$

Add 36 to 192.

$$x=\frac{-\left(-6\right)±\sqrt{228}}{2\times 6}$$

Take the square root of 228.

$$x=\frac{-\left(-6\right)±2\sqrt{57}}{2\times 6}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{57}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{6±2\sqrt{57}}{12}$$

Now solve the equation x=\frac{6±2\sqrt{57}}{12} when ± is plus. Add 6 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}+6}{12}$$

Divide 6+2\sqrt{57} by 12.

$$x=\frac{\sqrt{57}}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{6±2\sqrt{57}}{12} when ± is minus. Subtract 2\sqrt{57} from 6.

$$x=\frac{6-2\sqrt{57}}{12}$$

Divide 6-2\sqrt{57} by 12.

$$x=-\frac{\sqrt{57}}{6}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{\sqrt{57}}{6} for x_{1} and \frac{1}{2}-\frac{\sqrt{57}}{6} for x_{2}.

$$6x^{2}-6x-8=6\left(x-\left(\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)\left(x-\left(-\frac{\sqrt{57}}{6}+\frac{1}{2}\right)\right)$$