Solve -3x^2-7x+3

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

Factor

$$-3\left(x-\frac{-\sqrt{85}-7}{6}\right)\left(x-\frac{\sqrt{85}-7}{6}\right)$$

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.

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

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.

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 3}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 3}}{2\left(-3\right)}$$

Multiply 12 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49+36}}{2\left(-3\right)}$$

Add 49 to 36.

$$x=\frac{-\left(-7\right)±\sqrt{85}}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{85}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±\sqrt{85}}{-6}$$

Now solve the equation x=\frac{7±\sqrt{85}}{-6} when ± is plus. Add 7 to \sqrt{85}.

$$x=\frac{\sqrt{85}+7}{-6}$$

Divide 7+\sqrt{85} by -6.

$$x=\frac{-\sqrt{85}-7}{6}$$

Now solve the equation x=\frac{7±\sqrt{85}}{-6} when ± is minus. Subtract \sqrt{85} from 7.

$$x=\frac{7-\sqrt{85}}{-6}$$

Divide 7-\sqrt{85} by -6.

$$x=\frac{\sqrt{85}-7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7-\sqrt{85}}{6} for x_{1} and \frac{-7+\sqrt{85}}{6} for x_{2}.

$$-3x^{2}-7x+3=-3\left(x-\frac{-\sqrt{85}-7}{6}\right)\left(x-\frac{\sqrt{85}-7}{6}\right)$$

Evaluate

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