Solve 19-6x-9x^2

$$19 - 6 x - 9 x ^ { 2 }$$

Factor

$$-9\left(x-\frac{-2\sqrt{5}-1}{3}\right)\left(x-\frac{2\sqrt{5}-1}{3}\right)$$

Steps Using the Quadratic Formula

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.

$$-9x^{2}-6x+19=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-9\right)\times 19}}{2\left(-9\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-9\right)\times 19}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-6\right)±\sqrt{36+36\times 19}}{2\left(-9\right)}$$

Multiply 36 times 19.

$$x=\frac{-\left(-6\right)±\sqrt{36+684}}{2\left(-9\right)}$$

Add 36 to 684.

$$x=\frac{-\left(-6\right)±\sqrt{720}}{2\left(-9\right)}$$

Take the square root of 720.

$$x=\frac{-\left(-6\right)±12\sqrt{5}}{2\left(-9\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±12\sqrt{5}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{6±12\sqrt{5}}{-18}$$

Now solve the equation x=\frac{6±12\sqrt{5}}{-18} when ± is plus. Add 6 to 12\sqrt{5}.

$$x=\frac{12\sqrt{5}+6}{-18}$$

Divide 6+12\sqrt{5} by -18.

$$x=\frac{-2\sqrt{5}-1}{3}$$

Now solve the equation x=\frac{6±12\sqrt{5}}{-18} when ± is minus. Subtract 12\sqrt{5} from 6.

$$x=\frac{6-12\sqrt{5}}{-18}$$

Divide 6-12\sqrt{5} by -18.

$$x=\frac{2\sqrt{5}-1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-2\sqrt{5}}{3} for x_{1} and \frac{-1+2\sqrt{5}}{3} for x_{2}.

$$-9x^{2}-6x+19=-9\left(x-\frac{-2\sqrt{5}-1}{3}\right)\left(x-\frac{2\sqrt{5}-1}{3}\right)$$

Evaluate

$$19-6x-9x^{2}$$