Solve a^2+2a-15*4a-12

$${ a }^{ 2 } +2a-15 \cdot 4a-12$$

Evaluate

$$a^{2}-58a-12$$

Solution Steps

Multiply 15 and 4 to get 60.

$$a^{2}+2a-60a-12$$

Combine 2a and -60a to get -58a.

$$a^{2}-58a-12$$

Factor

$$\left(a-\left(29-\sqrt{853}\right)\right)\left(a-\left(\sqrt{853}+29\right)\right)$$

Steps Using the Quadratic Formula

Multiply 15 and 4 to get 60.

$$factor(a^{2}+2a-60a-12)$$

Combine 2a and -60a to get -58a.

$$factor(a^{2}-58a-12)$$

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.

$$a^{2}-58a-12=0$$

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.

$$a=\frac{-\left(-58\right)±\sqrt{\left(-58\right)^{2}-4\left(-12\right)}}{2}$$

Square -58.

$$a=\frac{-\left(-58\right)±\sqrt{3364-4\left(-12\right)}}{2}$$

Multiply -4 times -12.

$$a=\frac{-\left(-58\right)±\sqrt{3364+48}}{2}$$

Add 3364 to 48.

$$a=\frac{-\left(-58\right)±\sqrt{3412}}{2}$$

Take the square root of 3412.

$$a=\frac{-\left(-58\right)±2\sqrt{853}}{2}$$

The opposite of -58 is 58.

$$a=\frac{58±2\sqrt{853}}{2}$$

Now solve the equation a=\frac{58±2\sqrt{853}}{2} when ± is plus. Add 58 to 2\sqrt{853}.

$$a=\frac{2\sqrt{853}+58}{2}$$

Divide 58+2\sqrt{853} by 2.

$$a=\sqrt{853}+29$$

Now solve the equation a=\frac{58±2\sqrt{853}}{2} when ± is minus. Subtract 2\sqrt{853} from 58.

$$a=\frac{58-2\sqrt{853}}{2}$$

Divide 58-2\sqrt{853} by 2.

$$a=29-\sqrt{853}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 29+\sqrt{853} for x_{1} and 29-\sqrt{853} for x_{2}.

$$a^{2}-58a-12=\left(a-\left(\sqrt{853}+29\right)\right)\left(a-\left(29-\sqrt{853}\right)\right)$$