Solve (x^2+8x+8)+(-10x^2+5x)

$$( x ^ { 2 } + 8 x + 8 ) + ( - 10 x ^ { 2 } + 5 x )$$

Evaluate

$$8+13x-9x^{2}$$

Solution Steps

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$-9x^{2}+8x+8+5x$$

Combine 8x and 5x to get 13x.

$$-9x^{2}+13x+8$$

Factor

$$-9\left(x-\frac{13-\sqrt{457}}{18}\right)\left(x-\frac{\sqrt{457}+13}{18}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$factor(-9x^{2}+8x+8+5x)$$

Combine 8x and 5x to get 13x.

$$factor(-9x^{2}+13x+8)$$

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.

$$-9x^{2}+13x+8=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-13±\sqrt{169+36\times 8}}{2\left(-9\right)}$$

Multiply 36 times 8.

$$x=\frac{-13±\sqrt{169+288}}{2\left(-9\right)}$$

Add 169 to 288.

$$x=\frac{-13±\sqrt{457}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-13±\sqrt{457}}{-18}$$

Now solve the equation x=\frac{-13±\sqrt{457}}{-18} when ± is plus. Add -13 to \sqrt{457}.

$$x=\frac{\sqrt{457}-13}{-18}$$

Divide -13+\sqrt{457} by -18.

$$x=\frac{13-\sqrt{457}}{18}$$

Now solve the equation x=\frac{-13±\sqrt{457}}{-18} when ± is minus. Subtract \sqrt{457} from -13.

$$x=\frac{-\sqrt{457}-13}{-18}$$

Divide -13-\sqrt{457} by -18.

$$x=\frac{\sqrt{457}+13}{18}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13-\sqrt{457}}{18} for x_{1} and \frac{13+\sqrt{457}}{18} for x_{2}.

$$-9x^{2}+13x+8=-9\left(x-\frac{13-\sqrt{457}}{18}\right)\left(x-\frac{\sqrt{457}+13}{18}\right)$$