Solve 4a^2-9-12a

$$4 a ^ { 2 } - 9 - 12 a$$

Factor

$$4\left(a-\frac{3-3\sqrt{2}}{2}\right)\left(a-\frac{3\sqrt{2}+3}{2}\right)$$

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.

$$4a^{2}-12a-9=0$$

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.

$$a=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Square -12.

$$a=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-9\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-12\right)±\sqrt{144-16\left(-9\right)}}{2\times 4}$$

Multiply -16 times -9.

$$a=\frac{-\left(-12\right)±\sqrt{144+144}}{2\times 4}$$

Add 144 to 144.

$$a=\frac{-\left(-12\right)±\sqrt{288}}{2\times 4}$$

Take the square root of 288.

$$a=\frac{-\left(-12\right)±12\sqrt{2}}{2\times 4}$$

The opposite of -12 is 12.

$$a=\frac{12±12\sqrt{2}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{12±12\sqrt{2}}{8}$$

Now solve the equation a=\frac{12±12\sqrt{2}}{8} when ± is plus. Add 12 to 12\sqrt{2}.

$$a=\frac{12\sqrt{2}+12}{8}$$

Divide 12+12\sqrt{2} by 8.

$$a=\frac{3\sqrt{2}+3}{2}$$

Now solve the equation a=\frac{12±12\sqrt{2}}{8} when ± is minus. Subtract 12\sqrt{2} from 12.

$$a=\frac{12-12\sqrt{2}}{8}$$

Divide 12-12\sqrt{2} by 8.

$$a=\frac{3-3\sqrt{2}}{2}$$

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{2}}{2} for x_{1} and \frac{3-3\sqrt{2}}{2} for x_{2}.

$$4a^{2}-12a-9=4\left(a-\frac{3\sqrt{2}+3}{2}\right)\left(a-\frac{3-3\sqrt{2}}{2}\right)$$

Evaluate

$$4a^{2}-12a-9$$