Solve (x^2-6x-4)+(2x^2-5x+3)

$$( x ^ { 2 } - 6 x - 4 ) + ( 2 x ^ { 2 } - 5 x + 3 )$$

Evaluate

$$3x^{2}-11x-1$$

Solution Steps

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

$$3x^{2}-6x-4-5x+3$$

Combine -6x and -5x to get -11x.

$$3x^{2}-11x-4+3$$

Add -4 and 3 to get -1.

$$3x^{2}-11x-1$$

Factor

$$3\left(x-\frac{11-\sqrt{133}}{6}\right)\left(x-\frac{\sqrt{133}+11}{6}\right)$$

Steps Using the Quadratic Formula

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

$$factor(3x^{2}-6x-4-5x+3)$$

Combine -6x and -5x to get -11x.

$$factor(3x^{2}-11x-4+3)$$

Add -4 and 3 to get -1.

$$factor(3x^{2}-11x-1)$$

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.

$$3x^{2}-11x-1=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 3\left(-1\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-11\right)±\sqrt{121-12\left(-1\right)}}{2\times 3}$$

Multiply -12 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+12}}{2\times 3}$$

Add 121 to 12.

$$x=\frac{-\left(-11\right)±\sqrt{133}}{2\times 3}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{133}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{11±\sqrt{133}}{6}$$

Now solve the equation x=\frac{11±\sqrt{133}}{6} when ± is plus. Add 11 to \sqrt{133}.

$$x=\frac{\sqrt{133}+11}{6}$$

Now solve the equation x=\frac{11±\sqrt{133}}{6} when ± is minus. Subtract \sqrt{133} from 11.

$$x=\frac{11-\sqrt{133}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{133}}{6} for x_{1} and \frac{11-\sqrt{133}}{6} for x_{2}.

$$3x^{2}-11x-1=3\left(x-\frac{\sqrt{133}+11}{6}\right)\left(x-\frac{11-\sqrt{133}}{6}\right)$$