Solve 7-2x-3x^2

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

Factor

$$-3\left(x-\frac{-\sqrt{22}-1}{3}\right)\left(x-\frac{\sqrt{22}-1}{3}\right)$$

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.

$$-3x^{2}-2x+7=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-3\right)\times 7}}{2\left(-3\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-3\right)\times 7}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+12\times 7}}{2\left(-3\right)}$$

Multiply 12 times 7.

$$x=\frac{-\left(-2\right)±\sqrt{4+84}}{2\left(-3\right)}$$

Add 4 to 84.

$$x=\frac{-\left(-2\right)±\sqrt{88}}{2\left(-3\right)}$$

Take the square root of 88.

$$x=\frac{-\left(-2\right)±2\sqrt{22}}{2\left(-3\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{22}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{2±2\sqrt{22}}{-6}$$

Now solve the equation x=\frac{2±2\sqrt{22}}{-6} when ± is plus. Add 2 to 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+2}{-6}$$

Divide 2+2\sqrt{22} by -6.

$$x=\frac{-\sqrt{22}-1}{3}$$

Now solve the equation x=\frac{2±2\sqrt{22}}{-6} when ± is minus. Subtract 2\sqrt{22} from 2.

$$x=\frac{2-2\sqrt{22}}{-6}$$

Divide 2-2\sqrt{22} by -6.

$$x=\frac{\sqrt{22}-1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{22}}{3} for x_{1} and \frac{-1+\sqrt{22}}{3} for x_{2}.

$$-3x^{2}-2x+7=-3\left(x-\frac{-\sqrt{22}-1}{3}\right)\left(x-\frac{\sqrt{22}-1}{3}\right)$$

Evaluate

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