Solve (x^2-3x-3)+(2x^2+7x-2)

$$( x ^ { 2 } - 3 x - 3 ) + ( 2 x ^ { 2 } + 7 x - 2 )$$

Evaluate

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

Solution Steps

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

$$3x^{2}-3x-3+7x-2$$

Combine -3x and 7x to get 4x.

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

Subtract 2 from -3 to get -5.

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

Factor

$$3\left(x-\frac{-\sqrt{19}-2}{3}\right)\left(x-\frac{\sqrt{19}-2}{3}\right)$$

Steps Using the Quadratic Formula

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

$$factor(3x^{2}-3x-3+7x-2)$$

Combine -3x and 7x to get 4x.

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

Subtract 2 from -3 to get -5.

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

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.

$$3x^{2}+4x-5=0$$

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.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 3\left(-5\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-4±\sqrt{16-12\left(-5\right)}}{2\times 3}$$

Multiply -12 times -5.

$$x=\frac{-4±\sqrt{16+60}}{2\times 3}$$

Add 16 to 60.

$$x=\frac{-4±\sqrt{76}}{2\times 3}$$

Take the square root of 76.

$$x=\frac{-4±2\sqrt{19}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-4±2\sqrt{19}}{6}$$

Now solve the equation x=\frac{-4±2\sqrt{19}}{6} when ± is plus. Add -4 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-4}{6}$$

Divide -4+2\sqrt{19} by 6.

$$x=\frac{\sqrt{19}-2}{3}$$

Now solve the equation x=\frac{-4±2\sqrt{19}}{6} when ± is minus. Subtract 2\sqrt{19} from -4.

$$x=\frac{-2\sqrt{19}-4}{6}$$

Divide -4-2\sqrt{19} by 6.

$$x=\frac{-\sqrt{19}-2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-2+\sqrt{19}}{3} for x_{1} and \frac{-2-\sqrt{19}}{3} for x_{2}.

$$3x^{2}+4x-5=3\left(x-\frac{\sqrt{19}-2}{3}\right)\left(x-\frac{-\sqrt{19}-2}{3}\right)$$