Solve (4b^2-2)+(9b^2-3b

$$( 4 b ^ { 2 } - 2 ) + ( 9 b ^ { 2 } - 3 b$$

Factor

$$13\left(b-\frac{3-\sqrt{113}}{26}\right)\left(b-\frac{\sqrt{113}+3}{26}\right)$$

Steps Using the Quadratic Formula

Combine 4b^{2} and 9b^{2} to get 13b^{2}.

$$factor(13b^{2}-2-3b)$$

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.

$$13b^{2}-3b-2=0$$

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.

$$b=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 13\left(-2\right)}}{2\times 13}$$

Square -3.

$$b=\frac{-\left(-3\right)±\sqrt{9-4\times 13\left(-2\right)}}{2\times 13}$$

Multiply -4 times 13.

$$b=\frac{-\left(-3\right)±\sqrt{9-52\left(-2\right)}}{2\times 13}$$

Multiply -52 times -2.

$$b=\frac{-\left(-3\right)±\sqrt{9+104}}{2\times 13}$$

Add 9 to 104.

$$b=\frac{-\left(-3\right)±\sqrt{113}}{2\times 13}$$

The opposite of -3 is 3.

$$b=\frac{3±\sqrt{113}}{2\times 13}$$

Multiply 2 times 13.

$$b=\frac{3±\sqrt{113}}{26}$$

Now solve the equation b=\frac{3±\sqrt{113}}{26} when ± is plus. Add 3 to \sqrt{113}.

$$b=\frac{\sqrt{113}+3}{26}$$

Now solve the equation b=\frac{3±\sqrt{113}}{26} when ± is minus. Subtract \sqrt{113} from 3.

$$b=\frac{3-\sqrt{113}}{26}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{113}}{26} for x_{1} and \frac{3-\sqrt{113}}{26} for x_{2}.

$$13b^{2}-3b-2=13\left(b-\frac{\sqrt{113}+3}{26}\right)\left(b-\frac{3-\sqrt{113}}{26}\right)$$

Evaluate

$$13b^{2}-3b-2$$

Solution Steps

Combine 4b^{2} and 9b^{2} to get 13b^{2}.

$$13b^{2}-2-3b$$