Solve 2x^2+7x-2+(4x^2-x+1)

$$2 x ^ { 2 } + 7 x - 2 + ( 4 x ^ { 2 } - x + 1 )$$

Evaluate

$$6x^{2}+6x-1$$

Solution Steps

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}+7x-2-x+1$$

Combine 7x and -x to get 6x.

$$6x^{2}+6x-2+1$$

Add -2 and 1 to get -1.

$$6x^{2}+6x-1$$

Factor

$$6\left(x-\left(-\frac{\sqrt{15}}{6}-\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{15}}{6}-\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$factor(6x^{2}+7x-2-x+1)$$

Combine 7x and -x to get 6x.

$$factor(6x^{2}+6x-2+1)$$

Add -2 and 1 to get -1.

$$factor(6x^{2}+6x-1)$$

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.

$$6x^{2}+6x-1=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-6±\sqrt{36-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$x=\frac{-6±\sqrt{36+24}}{2\times 6}$$

Add 36 to 24.

$$x=\frac{-6±\sqrt{60}}{2\times 6}$$

Take the square root of 60.

$$x=\frac{-6±2\sqrt{15}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-6±2\sqrt{15}}{12}$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{12} when ± is plus. Add -6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-6}{12}$$

Divide -6+2\sqrt{15} by 12.

$$x=\frac{\sqrt{15}}{6}-\frac{1}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{12} when ± is minus. Subtract 2\sqrt{15} from -6.

$$x=\frac{-2\sqrt{15}-6}{12}$$

Divide -6-2\sqrt{15} by 12.

$$x=-\frac{\sqrt{15}}{6}-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2}+\frac{\sqrt{15}}{6} for x_{1} and -\frac{1}{2}-\frac{\sqrt{15}}{6} for x_{2}.

$$6x^{2}+6x-1=6\left(x-\left(\frac{\sqrt{15}}{6}-\frac{1}{2}\right)\right)\left(x-\left(-\frac{\sqrt{15}}{6}-\frac{1}{2}\right)\right)$$