Solve 7-4x-x^2

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

Factor

$$-\left(x-\left(-\sqrt{11}-2\right)\right)\left(x-\left(\sqrt{11}-2\right)\right)$$

Steps Using the Quadratic Formula

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.

$$-x^{2}-4x+7=0$$

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$x=\frac{-\left(-4\right)±\sqrt{16+28}}{2\left(-1\right)}$$

Add 16 to 28.

$$x=\frac{-\left(-4\right)±\sqrt{44}}{2\left(-1\right)}$$

Take the square root of 44.

$$x=\frac{-\left(-4\right)±2\sqrt{11}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{11}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±2\sqrt{11}}{-2}$$

Now solve the equation x=\frac{4±2\sqrt{11}}{-2} when ± is plus. Add 4 to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}+4}{-2}$$

Divide 4+2\sqrt{11} by -2.

$$x=-\left(\sqrt{11}+2\right)$$

Now solve the equation x=\frac{4±2\sqrt{11}}{-2} when ± is minus. Subtract 2\sqrt{11} from 4.

$$x=\frac{4-2\sqrt{11}}{-2}$$

Divide 4-2\sqrt{11} by -2.

$$x=\sqrt{11}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(2+\sqrt{11}\right) for x_{1} and -2+\sqrt{11} for x_{2}.

$$-x^{2}-4x+7=-\left(x-\left(-\left(\sqrt{11}+2\right)\right)\right)\left(x-\left(\sqrt{11}-2\right)\right)$$

Evaluate

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