Solve x^2+x-5-3x^2-7x+14

$$x ^ { 2 } + x - 5 - 3 x ^ { 2 } - 7 x + 14$$

Evaluate

$$9-6x-2x^{2}$$

Solution Steps

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

$$-2x^{2}+x-5-7x+14$$

Combine x and -7x to get -6x.

$$-2x^{2}-6x-5+14$$

Add -5 and 14 to get 9.

$$-2x^{2}-6x+9$$

Factor

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

Steps Using the Quadratic Formula

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

$$factor(-2x^{2}+x-5-7x+14)$$

Combine x and -7x to get -6x.

$$factor(-2x^{2}-6x-5+14)$$

Add -5 and 14 to get 9.

$$factor(-2x^{2}-6x+9)$$

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.

$$-2x^{2}-6x+9=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-6\right)±\sqrt{36+8\times 9}}{2\left(-2\right)}$$

Multiply 8 times 9.

$$x=\frac{-\left(-6\right)±\sqrt{36+72}}{2\left(-2\right)}$$

Add 36 to 72.

$$x=\frac{-\left(-6\right)±\sqrt{108}}{2\left(-2\right)}$$

Take the square root of 108.

$$x=\frac{-\left(-6\right)±6\sqrt{3}}{2\left(-2\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±6\sqrt{3}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{6±6\sqrt{3}}{-4}$$

Now solve the equation x=\frac{6±6\sqrt{3}}{-4} when ± is plus. Add 6 to 6\sqrt{3}.

$$x=\frac{6\sqrt{3}+6}{-4}$$

Divide 6+6\sqrt{3} by -4.

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

Now solve the equation x=\frac{6±6\sqrt{3}}{-4} when ± is minus. Subtract 6\sqrt{3} from 6.

$$x=\frac{6-6\sqrt{3}}{-4}$$

Divide 6-6\sqrt{3} by -4.

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

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

$$-2x^{2}-6x+9=-2\left(x-\frac{-3\sqrt{3}-3}{2}\right)\left(x-\frac{3\sqrt{3}-3}{2}\right)$$