Solve F(x)=x^2+2x^2-5x-6

$$F ( x ) = x ^ { 2 } + 2 x ^ { 2 } - 5 x - 6$$

Factor

$$3\left(x-\frac{5-\sqrt{97}}{6}\right)\left(x-\frac{\sqrt{97}+5}{6}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$factor(3x^{2}-5x-6)$$

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.

$$3x^{2}-5x-6=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-6\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-6\right)}}{2\times 3}$$

Multiply -12 times -6.

$$x=\frac{-\left(-5\right)±\sqrt{25+72}}{2\times 3}$$

Add 25 to 72.

$$x=\frac{-\left(-5\right)±\sqrt{97}}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{97}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±\sqrt{97}}{6}$$

Now solve the equation x=\frac{5±\sqrt{97}}{6} when ± is plus. Add 5 to \sqrt{97}.

$$x=\frac{\sqrt{97}+5}{6}$$

Now solve the equation x=\frac{5±\sqrt{97}}{6} when ± is minus. Subtract \sqrt{97} from 5.

$$x=\frac{5-\sqrt{97}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{97}}{6} for x_{1} and \frac{5-\sqrt{97}}{6} for x_{2}.

$$3x^{2}-5x-6=3\left(x-\frac{\sqrt{97}+5}{6}\right)\left(x-\frac{5-\sqrt{97}}{6}\right)$$

Evaluate

$$3x^{2}-5x-6$$

Solution Steps

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$3x^{2}-5x-6$$