Solve (-9c^2-5c+7)+(3c+9)

$$( - 9 c ^ { 2 } - 5 c + 7 ) + ( 3 c + 9 )$$

Evaluate

$$16-2c-9c^{2}$$

Solution Steps

Combine -5c and 3c to get -2c.

$$-9c^{2}-2c+7+9$$

Add 7 and 9 to get 16.

$$-9c^{2}-2c+16$$

Factor

$$-9\left(c-\frac{-\sqrt{145}-1}{9}\right)\left(c-\frac{\sqrt{145}-1}{9}\right)$$

Steps Using the Quadratic Formula

Combine -5c and 3c to get -2c.

$$factor(-9c^{2}-2c+7+9)$$

Add 7 and 9 to get 16.

$$factor(-9c^{2}-2c+16)$$

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.

$$-9c^{2}-2c+16=0$$

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.

$$c=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-9\right)\times 16}}{2\left(-9\right)}$$

Square -2.

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

Multiply -4 times -9.

$$c=\frac{-\left(-2\right)±\sqrt{4+36\times 16}}{2\left(-9\right)}$$

Multiply 36 times 16.

$$c=\frac{-\left(-2\right)±\sqrt{4+576}}{2\left(-9\right)}$$

Add 4 to 576.

$$c=\frac{-\left(-2\right)±\sqrt{580}}{2\left(-9\right)}$$

Take the square root of 580.

$$c=\frac{-\left(-2\right)±2\sqrt{145}}{2\left(-9\right)}$$

The opposite of -2 is 2.

$$c=\frac{2±2\sqrt{145}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$c=\frac{2±2\sqrt{145}}{-18}$$

Now solve the equation c=\frac{2±2\sqrt{145}}{-18} when ± is plus. Add 2 to 2\sqrt{145}.

$$c=\frac{2\sqrt{145}+2}{-18}$$

Divide 2+2\sqrt{145} by -18.

$$c=\frac{-\sqrt{145}-1}{9}$$

Now solve the equation c=\frac{2±2\sqrt{145}}{-18} when ± is minus. Subtract 2\sqrt{145} from 2.

$$c=\frac{2-2\sqrt{145}}{-18}$$

Divide 2-2\sqrt{145} by -18.

$$c=\frac{\sqrt{145}-1}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{145}}{9} for x_{1} and \frac{-1+\sqrt{145}}{9} for x_{2}.

$$-9c^{2}-2c+16=-9\left(c-\frac{-\sqrt{145}-1}{9}\right)\left(c-\frac{\sqrt{145}-1}{9}\right)$$