Solve (6a^2+a-18)+(12a+5a^2+8a)

$$( 6 a ^ { 2 } + a - 18 ) + ( 12 a + 5 a ^ { 2 } + 8 a )$$

Evaluate

$$11a^{2}+21a-18$$

Solution Steps

Combine a and 12a to get 13a.

$$6a^{2}+13a-18+5a^{2}+8a$$

Combine 6a^{2} and 5a^{2} to get 11a^{2}.

$$11a^{2}+13a-18+8a$$

Combine 13a and 8a to get 21a.

$$11a^{2}+21a-18$$

Factor

$$11\left(a-\frac{-3\sqrt{137}-21}{22}\right)\left(a-\frac{3\sqrt{137}-21}{22}\right)$$

Steps Using the Quadratic Formula

Combine a and 12a to get 13a.

$$factor(6a^{2}+13a-18+5a^{2}+8a)$$

Combine 6a^{2} and 5a^{2} to get 11a^{2}.

$$factor(11a^{2}+13a-18+8a)$$

Combine 13a and 8a to get 21a.

$$factor(11a^{2}+21a-18)$$

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.

$$11a^{2}+21a-18=0$$

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.

$$a=\frac{-21±\sqrt{21^{2}-4\times 11\left(-18\right)}}{2\times 11}$$

Square 21.

$$a=\frac{-21±\sqrt{441-4\times 11\left(-18\right)}}{2\times 11}$$

Multiply -4 times 11.

$$a=\frac{-21±\sqrt{441-44\left(-18\right)}}{2\times 11}$$

Multiply -44 times -18.

$$a=\frac{-21±\sqrt{441+792}}{2\times 11}$$

Add 441 to 792.

$$a=\frac{-21±\sqrt{1233}}{2\times 11}$$

Take the square root of 1233.

$$a=\frac{-21±3\sqrt{137}}{2\times 11}$$

Multiply 2 times 11.

$$a=\frac{-21±3\sqrt{137}}{22}$$

Now solve the equation a=\frac{-21±3\sqrt{137}}{22} when ± is plus. Add -21 to 3\sqrt{137}.

$$a=\frac{3\sqrt{137}-21}{22}$$

Now solve the equation a=\frac{-21±3\sqrt{137}}{22} when ± is minus. Subtract 3\sqrt{137} from -21.

$$a=\frac{-3\sqrt{137}-21}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-21+3\sqrt{137}}{22} for x_{1} and \frac{-21-3\sqrt{137}}{22} for x_{2}.

$$11a^{2}+21a-18=11\left(a-\frac{3\sqrt{137}-21}{22}\right)\left(a-\frac{-3\sqrt{137}-21}{22}\right)$$