Solve (-9f^2+9f+6)+(-7f+4)

$$( - 9 f ^ { 2 } + 9 f + 6 ) + ( - 7 f + 4 )$$

Evaluate

$$10+2f-9f^{2}$$

Solution Steps

Combine 9f and -7f to get 2f.

$$-9f^{2}+2f+6+4$$

Add 6 and 4 to get 10.

$$-9f^{2}+2f+10$$

Factor

$$-9\left(f-\frac{1-\sqrt{91}}{9}\right)\left(f-\frac{\sqrt{91}+1}{9}\right)$$

Steps Using the Quadratic Formula

Combine 9f and -7f to get 2f.

$$factor(-9f^{2}+2f+6+4)$$

Add 6 and 4 to get 10.

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

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.

$$-9f^{2}+2f+10=0$$

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.

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

Square 2.

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

Multiply -4 times -9.

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

Multiply 36 times 10.

$$f=\frac{-2±\sqrt{4+360}}{2\left(-9\right)}$$

Add 4 to 360.

$$f=\frac{-2±\sqrt{364}}{2\left(-9\right)}$$

Take the square root of 364.

$$f=\frac{-2±2\sqrt{91}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$f=\frac{-2±2\sqrt{91}}{-18}$$

Now solve the equation f=\frac{-2±2\sqrt{91}}{-18} when ± is plus. Add -2 to 2\sqrt{91}.

$$f=\frac{2\sqrt{91}-2}{-18}$$

Divide -2+2\sqrt{91} by -18.

$$f=\frac{1-\sqrt{91}}{9}$$

Now solve the equation f=\frac{-2±2\sqrt{91}}{-18} when ± is minus. Subtract 2\sqrt{91} from -2.

$$f=\frac{-2\sqrt{91}-2}{-18}$$

Divide -2-2\sqrt{91} by -18.

$$f=\frac{\sqrt{91}+1}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{91}}{9} for x_{1} and \frac{1+\sqrt{91}}{9} for x_{2}.

$$-9f^{2}+2f+10=-9\left(f-\frac{1-\sqrt{91}}{9}\right)\left(f-\frac{\sqrt{91}+1}{9}\right)$$