Solve -7x+4x^2-7+x-5x^2

$$-7x+4 { x }^{ 2 } -7+x-5 { x }^{ 2 }$$

Evaluate

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

Solution Steps

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

$$-6x+4x^{2}-7-5x^{2}$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

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

Factor

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

Steps Using the Quadratic Formula

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

$$factor(-6x+4x^{2}-7-5x^{2})$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$factor(-6x-x^{2}-7)$$

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.

$$-x^{2}-6x-7=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$x=\frac{-\left(-6\right)±\sqrt{36-28}}{2\left(-1\right)}$$

Add 36 to -28.

$$x=\frac{-\left(-6\right)±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

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

The opposite of -6 is 6.

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

Multiply 2 times -1.

$$x=\frac{6±2\sqrt{2}}{-2}$$

Now solve the equation x=\frac{6±2\sqrt{2}}{-2} when ± is plus. Add 6 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+6}{-2}$$

Divide 6+2\sqrt{2} by -2.

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

Now solve the equation x=\frac{6±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from 6.

$$x=\frac{6-2\sqrt{2}}{-2}$$

Divide 6-2\sqrt{2} by -2.

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

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

$$-x^{2}-6x-7=-\left(x-\left(-\left(\sqrt{2}+3\right)\right)\right)\left(x-\left(\sqrt{2}-3\right)\right)$$