Solve 6x^2-7x^2-5x+3x+9-3

$$6 { x }^{ 2 } -7 { x }^{ 2 } -5x+3x+9-3$$

Evaluate

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

Solution Steps

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

$$-x^{2}-5x+3x+9-3$$

Combine -5x and 3x to get -2x.

$$-x^{2}-2x+9-3$$

Subtract 3 from 9 to get 6.

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

Factor

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

Steps Using the Quadratic Formula

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

$$factor(-x^{2}-5x+3x+9-3)$$

Combine -5x and 3x to get -2x.

$$factor(-x^{2}-2x+9-3)$$

Subtract 3 from 9 to get 6.

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

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.

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

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.

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 6.

$$x=\frac{-\left(-2\right)±\sqrt{4+24}}{2\left(-1\right)}$$

Add 4 to 24.

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

Take the square root of 28.

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

The opposite of -2 is 2.

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

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{7}}{-2}$$

Now solve the equation x=\frac{2±2\sqrt{7}}{-2} when ± is plus. Add 2 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+2}{-2}$$

Divide 2+2\sqrt{7} by -2.

$$x=-\left(\sqrt{7}+1\right)$$

Now solve the equation x=\frac{2±2\sqrt{7}}{-2} when ± is minus. Subtract 2\sqrt{7} from 2.

$$x=\frac{2-2\sqrt{7}}{-2}$$

Divide 2-2\sqrt{7} by -2.

$$x=\sqrt{7}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(1+\sqrt{7}\right) for x_{1} and -1+\sqrt{7} for x_{2}.

$$-x^{2}-2x+6=-\left(x-\left(-\left(\sqrt{7}+1\right)\right)\right)\left(x-\left(\sqrt{7}-1\right)\right)$$