Solve (-2f^2-3f-5)+(-2f^2-3f+8)

$$( - 2 f ^ { 2 } - 3 f - 5 ) + ( - 2 f ^ { 2 } - 3 f + 8 )$$

Evaluate

$$3-6f-4f^{2}$$

Solution Steps

Combine -2f^{2} and -2f^{2} to get -4f^{2}.

$$-4f^{2}-3f-5-3f+8$$

Combine -3f and -3f to get -6f.

$$-4f^{2}-6f-5+8$$

Add -5 and 8 to get 3.

$$-4f^{2}-6f+3$$

Factor

$$-4\left(f-\frac{-\sqrt{21}-3}{4}\right)\left(f-\frac{\sqrt{21}-3}{4}\right)$$

Steps Using the Quadratic Formula

Combine -2f^{2} and -2f^{2} to get -4f^{2}.

$$factor(-4f^{2}-3f-5-3f+8)$$

Combine -3f and -3f to get -6f.

$$factor(-4f^{2}-6f-5+8)$$

Add -5 and 8 to get 3.

$$factor(-4f^{2}-6f+3)$$

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.

$$-4f^{2}-6f+3=0$$

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.

$$f=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Square -6.

$$f=\frac{-\left(-6\right)±\sqrt{36-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$f=\frac{-\left(-6\right)±\sqrt{36+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$f=\frac{-\left(-6\right)±\sqrt{36+48}}{2\left(-4\right)}$$

Add 36 to 48.

$$f=\frac{-\left(-6\right)±\sqrt{84}}{2\left(-4\right)}$$

Take the square root of 84.

$$f=\frac{-\left(-6\right)±2\sqrt{21}}{2\left(-4\right)}$$

The opposite of -6 is 6.

$$f=\frac{6±2\sqrt{21}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$f=\frac{6±2\sqrt{21}}{-8}$$

Now solve the equation f=\frac{6±2\sqrt{21}}{-8} when ± is plus. Add 6 to 2\sqrt{21}.

$$f=\frac{2\sqrt{21}+6}{-8}$$

Divide 6+2\sqrt{21} by -8.

$$f=\frac{-\sqrt{21}-3}{4}$$

Now solve the equation f=\frac{6±2\sqrt{21}}{-8} when ± is minus. Subtract 2\sqrt{21} from 6.

$$f=\frac{6-2\sqrt{21}}{-8}$$

Divide 6-2\sqrt{21} by -8.

$$f=\frac{\sqrt{21}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3-\sqrt{21}}{4} for x_{1} and \frac{-3+\sqrt{21}}{4} for x_{2}.

$$-4f^{2}-6f+3=-4\left(f-\frac{-\sqrt{21}-3}{4}\right)\left(f-\frac{\sqrt{21}-3}{4}\right)$$