Solve 6x^2-3x-7x^2-5x-2

$$6 x ^ { 2 } - 3 x - 7 x ^ { 2 } - 5 x - 2$$

Evaluate

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

Solution Steps

Combine 6x^{2} and -7x^{2} to get -x^{2}.

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

Combine -3x and -5x to get -8x.

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

Factor

$$-\left(x-\left(-\sqrt{14}-4\right)\right)\left(x-\left(\sqrt{14}-4\right)\right)$$

Steps Using the Quadratic Formula

Combine 6x^{2} and -7x^{2} to get -x^{2}.

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

Combine -3x and -5x to get -8x.

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

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.

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

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.

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

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8}}{2\left(-1\right)}$$

Add 64 to -8.

$$x=\frac{-\left(-8\right)±\sqrt{56}}{2\left(-1\right)}$$

Take the square root of 56.

$$x=\frac{-\left(-8\right)±2\sqrt{14}}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{14}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±2\sqrt{14}}{-2}$$

Now solve the equation x=\frac{8±2\sqrt{14}}{-2} when ± is plus. Add 8 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}+8}{-2}$$

Divide 8+2\sqrt{14} by -2.

$$x=-\left(\sqrt{14}+4\right)$$

Now solve the equation x=\frac{8±2\sqrt{14}}{-2} when ± is minus. Subtract 2\sqrt{14} from 8.

$$x=\frac{8-2\sqrt{14}}{-2}$$

Divide 8-2\sqrt{14} by -2.

$$x=\sqrt{14}-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(4+\sqrt{14}\right) for x_{1} and -4+\sqrt{14} for x_{2}.

$$-x^{2}-8x-2=-\left(x-\left(-\left(\sqrt{14}+4\right)\right)\right)\left(x-\left(\sqrt{14}-4\right)\right)$$