Solve (3x^2+4x-12)+(2x^2-6x+10)

$$( 3 x ^ { 2 } + 4 x - 12 ) + ( 2 x ^ { 2 } - 6 x + 10 )$$

Evaluate

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

Solution Steps

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}+4x-12-6x+10$$

Combine 4x and -6x to get -2x.

$$5x^{2}-2x-12+10$$

Add -12 and 10 to get -2.

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

Factor

$$5\left(x-\frac{1-\sqrt{11}}{5}\right)\left(x-\frac{\sqrt{11}+1}{5}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$factor(5x^{2}+4x-12-6x+10)$$

Combine 4x and -6x to get -2x.

$$factor(5x^{2}-2x-12+10)$$

Add -12 and 10 to get -2.

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

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.

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

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.

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

Square -2.

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

Multiply -4 times 5.

$$x=\frac{-\left(-2\right)±\sqrt{4-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$x=\frac{-\left(-2\right)±\sqrt{4+40}}{2\times 5}$$

Add 4 to 40.

$$x=\frac{-\left(-2\right)±\sqrt{44}}{2\times 5}$$

Take the square root of 44.

$$x=\frac{-\left(-2\right)±2\sqrt{11}}{2\times 5}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{11}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{2±2\sqrt{11}}{10}$$

Now solve the equation x=\frac{2±2\sqrt{11}}{10} when ± is plus. Add 2 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+2}{10}$$

Divide 2+2\sqrt{11} by 10.

$$x=\frac{\sqrt{11}+1}{5}$$

Now solve the equation x=\frac{2±2\sqrt{11}}{10} when ± is minus. Subtract 2\sqrt{11} from 2.

$$x=\frac{2-2\sqrt{11}}{10}$$

Divide 2-2\sqrt{11} by 10.

$$x=\frac{1-\sqrt{11}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{11}}{5} for x_{1} and \frac{1-\sqrt{11}}{5} for x_{2}.

$$5x^{2}-2x-2=5\left(x-\frac{\sqrt{11}+1}{5}\right)\left(x-\frac{1-\sqrt{11}}{5}\right)$$