Solve a^2+6a+9a^2-9

$$a ^ { 2 } + 6 a + 9 a ^ { 2 } - 9$$

Factor

$$10\left(a-\frac{-3\sqrt{11}-3}{10}\right)\left(a-\frac{3\sqrt{11}-3}{10}\right)$$

Steps Using the Quadratic Formula

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$factor(10a^{2}+6a-9)$$

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.

$$10a^{2}+6a-9=0$$

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.

$$a=\frac{-6±\sqrt{6^{2}-4\times 10\left(-9\right)}}{2\times 10}$$

Square 6.

$$a=\frac{-6±\sqrt{36-4\times 10\left(-9\right)}}{2\times 10}$$

Multiply -4 times 10.

$$a=\frac{-6±\sqrt{36-40\left(-9\right)}}{2\times 10}$$

Multiply -40 times -9.

$$a=\frac{-6±\sqrt{36+360}}{2\times 10}$$

Add 36 to 360.

$$a=\frac{-6±\sqrt{396}}{2\times 10}$$

Take the square root of 396.

$$a=\frac{-6±6\sqrt{11}}{2\times 10}$$

Multiply 2 times 10.

$$a=\frac{-6±6\sqrt{11}}{20}$$

Now solve the equation a=\frac{-6±6\sqrt{11}}{20} when ± is plus. Add -6 to 6\sqrt{11}.

$$a=\frac{6\sqrt{11}-6}{20}$$

Divide -6+6\sqrt{11} by 20.

$$a=\frac{3\sqrt{11}-3}{10}$$

Now solve the equation a=\frac{-6±6\sqrt{11}}{20} when ± is minus. Subtract 6\sqrt{11} from -6.

$$a=\frac{-6\sqrt{11}-6}{20}$$

Divide -6-6\sqrt{11} by 20.

$$a=\frac{-3\sqrt{11}-3}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+3\sqrt{11}}{10} for x_{1} and \frac{-3-3\sqrt{11}}{10} for x_{2}.

$$10a^{2}+6a-9=10\left(a-\frac{3\sqrt{11}-3}{10}\right)\left(a-\frac{-3\sqrt{11}-3}{10}\right)$$

Evaluate

$$10a^{2}+6a-9$$

Solution Steps

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$10a^{2}+6a-9$$