Solve (-5x^2-4x-1)+(-4x^2-4x)

$$( - 5 x ^ { 2 } - 4 x - 1 ) + ( - 4 x ^ { 2 } - 4 x )$$

Evaluate

$$-9x^{2}-8x-1$$

Solution Steps

Combine -5x^{2} and -4x^{2} to get -9x^{2}.

$$-9x^{2}-4x-1-4x$$

Combine -4x and -4x to get -8x.

$$-9x^{2}-8x-1$$

Factor

$$-9\left(x-\frac{-\sqrt{7}-4}{9}\right)\left(x-\frac{\sqrt{7}-4}{9}\right)$$

Steps Using the Quadratic Formula

Combine -5x^{2} and -4x^{2} to get -9x^{2}.

$$factor(-9x^{2}-4x-1-4x)$$

Combine -4x and -4x to get -8x.

$$factor(-9x^{2}-8x-1)$$

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.

$$-9x^{2}-8x-1=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-9\right)\left(-1\right)}}{2\left(-9\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)\left(-1\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-8\right)±\sqrt{64+36\left(-1\right)}}{2\left(-9\right)}$$

Multiply 36 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64-36}}{2\left(-9\right)}$$

Add 64 to -36.

$$x=\frac{-\left(-8\right)±\sqrt{28}}{2\left(-9\right)}$$

Take the square root of 28.

$$x=\frac{-\left(-8\right)±2\sqrt{7}}{2\left(-9\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{7}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{8±2\sqrt{7}}{-18}$$

Now solve the equation x=\frac{8±2\sqrt{7}}{-18} when ± is plus. Add 8 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+8}{-18}$$

Divide 8+2\sqrt{7} by -18.

$$x=\frac{-\sqrt{7}-4}{9}$$

Now solve the equation x=\frac{8±2\sqrt{7}}{-18} when ± is minus. Subtract 2\sqrt{7} from 8.

$$x=\frac{8-2\sqrt{7}}{-18}$$

Divide 8-2\sqrt{7} by -18.

$$x=\frac{\sqrt{7}-4}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-4-\sqrt{7}}{9} for x_{1} and \frac{-4+\sqrt{7}}{9} for x_{2}.

$$-9x^{2}-8x-1=-9\left(x-\frac{-\sqrt{7}-4}{9}\right)\left(x-\frac{\sqrt{7}-4}{9}\right)$$