Solve (-3x^2+x)+(-4x^2-2x+7)

$$( - 3 x ^ { 2 } + x ) + ( - 4 x ^ { 2 } - 2 x + 7 )$$

Evaluate

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

Solution Steps

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

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

Combine x and -2x to get -x.

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

Factor

$$-7\left(x-\frac{-\sqrt{197}-1}{14}\right)\left(x-\frac{\sqrt{197}-1}{14}\right)$$

Steps Using the Quadratic Formula

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

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

Combine x and -2x to get -x.

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

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.

$$-7x^{2}-x+7=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-1\right)±\sqrt{1+28\times 7}}{2\left(-7\right)}$$

Multiply 28 times 7.

$$x=\frac{-\left(-1\right)±\sqrt{1+196}}{2\left(-7\right)}$$

Add 1 to 196.

$$x=\frac{-\left(-1\right)±\sqrt{197}}{2\left(-7\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{197}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{1±\sqrt{197}}{-14}$$

Now solve the equation x=\frac{1±\sqrt{197}}{-14} when ± is plus. Add 1 to \sqrt{197}.

$$x=\frac{\sqrt{197}+1}{-14}$$

Divide 1+\sqrt{197} by -14.

$$x=\frac{-\sqrt{197}-1}{14}$$

Now solve the equation x=\frac{1±\sqrt{197}}{-14} when ± is minus. Subtract \sqrt{197} from 1.

$$x=\frac{1-\sqrt{197}}{-14}$$

Divide 1-\sqrt{197} by -14.

$$x=\frac{\sqrt{197}-1}{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{197}}{14} for x_{1} and \frac{-1+\sqrt{197}}{14} for x_{2}.

$$-7x^{2}-x+7=-7\left(x-\frac{-\sqrt{197}-1}{14}\right)\left(x-\frac{\sqrt{197}-1}{14}\right)$$