Solve -5x^2+67x-30

$$-5 { x }^{ 2 } +67x-30$$

Factor

$$-5\left(x-\frac{67-\sqrt{3889}}{10}\right)\left(x-\frac{\sqrt{3889}+67}{10}\right)$$

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.

$$-5x^{2}+67x-30=0$$

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.

$$x=\frac{-67±\sqrt{67^{2}-4\left(-5\right)\left(-30\right)}}{2\left(-5\right)}$$

Square 67.

$$x=\frac{-67±\sqrt{4489-4\left(-5\right)\left(-30\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-67±\sqrt{4489+20\left(-30\right)}}{2\left(-5\right)}$$

Multiply 20 times -30.

$$x=\frac{-67±\sqrt{4489-600}}{2\left(-5\right)}$$

Add 4489 to -600.

$$x=\frac{-67±\sqrt{3889}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-67±\sqrt{3889}}{-10}$$

Now solve the equation x=\frac{-67±\sqrt{3889}}{-10} when ± is plus. Add -67 to \sqrt{3889}.

$$x=\frac{\sqrt{3889}-67}{-10}$$

Divide -67+\sqrt{3889} by -10.

$$x=\frac{67-\sqrt{3889}}{10}$$

Now solve the equation x=\frac{-67±\sqrt{3889}}{-10} when ± is minus. Subtract \sqrt{3889} from -67.

$$x=\frac{-\sqrt{3889}-67}{-10}$$

Divide -67-\sqrt{3889} by -10.

$$x=\frac{\sqrt{3889}+67}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{67-\sqrt{3889}}{10} for x_{1} and \frac{67+\sqrt{3889}}{10} for x_{2}.

$$-5x^{2}+67x-30=-5\left(x-\frac{67-\sqrt{3889}}{10}\right)\left(x-\frac{\sqrt{3889}+67}{10}\right)$$

Evaluate

$$-5x^{2}+67x-30$$