Solve (-3x^2+2x+1)+(-7x^2-2x-3)+(2x^2-x+13)=

$$( - 3 x ^ { 2 } + 2 x + 1 ) + ( - 7 x ^ { 2 } - 2 x - 3 ) + ( 2 x ^ { 2 } - x + 13 ) =$$

Evaluate

$$11-x-8x^{2}$$

Solution Steps

Combine -3x^{2} and -7x^{2} to get -10x^{2}.

$$-10x^{2}+2x+1-2x-3+2x^{2}-x+13$$

Combine 2x and -2x to get 0.

$$-10x^{2}+1-3+2x^{2}-x+13$$

Subtract 3 from 1 to get -2.

$$-10x^{2}-2+2x^{2}-x+13$$

Combine -10x^{2} and 2x^{2} to get -8x^{2}.

$$-8x^{2}-2-x+13$$

Add -2 and 13 to get 11.

$$-8x^{2}+11-x$$

Factor

$$-8\left(x-\frac{-\sqrt{353}-1}{16}\right)\left(x-\frac{\sqrt{353}-1}{16}\right)$$

Steps Using the Quadratic Formula

Combine -3x^{2} and -7x^{2} to get -10x^{2}.

$$factor(-10x^{2}+2x+1-2x-3+2x^{2}-x+13)$$

Combine 2x and -2x to get 0.

$$factor(-10x^{2}+1-3+2x^{2}-x+13)$$

Subtract 3 from 1 to get -2.

$$factor(-10x^{2}-2+2x^{2}-x+13)$$

Combine -10x^{2} and 2x^{2} to get -8x^{2}.

$$factor(-8x^{2}-2-x+13)$$

Add -2 and 13 to get 11.

$$factor(-8x^{2}+11-x)$$

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.

$$-8x^{2}-x+11=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-8\right)\times 11}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-1\right)±\sqrt{1+32\times 11}}{2\left(-8\right)}$$

Multiply 32 times 11.

$$x=\frac{-\left(-1\right)±\sqrt{1+352}}{2\left(-8\right)}$$

Add 1 to 352.

$$x=\frac{-\left(-1\right)±\sqrt{353}}{2\left(-8\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{353}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{1±\sqrt{353}}{-16}$$

Now solve the equation x=\frac{1±\sqrt{353}}{-16} when ± is plus. Add 1 to \sqrt{353}.

$$x=\frac{\sqrt{353}+1}{-16}$$

Divide 1+\sqrt{353} by -16.

$$x=\frac{-\sqrt{353}-1}{16}$$

Now solve the equation x=\frac{1±\sqrt{353}}{-16} when ± is minus. Subtract \sqrt{353} from 1.

$$x=\frac{1-\sqrt{353}}{-16}$$

Divide 1-\sqrt{353} by -16.

$$x=\frac{\sqrt{353}-1}{16}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{353}}{16} for x_{1} and \frac{-1+\sqrt{353}}{16} for x_{2}.

$$-8x^{2}-x+11=-8\left(x-\frac{-\sqrt{353}-1}{16}\right)\left(x-\frac{\sqrt{353}-1}{16}\right)$$