Solve -7x^2+28x-10

$$-7 { x }^{ 2 } +28x-10$$

Factor

$$-7\left(x-\left(-\frac{3\sqrt{14}}{7}+2\right)\right)\left(x-\left(\frac{3\sqrt{14}}{7}+2\right)\right)$$

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.

$$-7x^{2}+28x-10=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-7\right)\left(-10\right)}}{2\left(-7\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-7\right)\left(-10\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-28±\sqrt{784+28\left(-10\right)}}{2\left(-7\right)}$$

Multiply 28 times -10.

$$x=\frac{-28±\sqrt{784-280}}{2\left(-7\right)}$$

Add 784 to -280.

$$x=\frac{-28±\sqrt{504}}{2\left(-7\right)}$$

Take the square root of 504.

$$x=\frac{-28±6\sqrt{14}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-28±6\sqrt{14}}{-14}$$

Now solve the equation x=\frac{-28±6\sqrt{14}}{-14} when ± is plus. Add -28 to 6\sqrt{14}.

$$x=\frac{6\sqrt{14}-28}{-14}$$

Divide -28+6\sqrt{14} by -14.

$$x=-\frac{3\sqrt{14}}{7}+2$$

Now solve the equation x=\frac{-28±6\sqrt{14}}{-14} when ± is minus. Subtract 6\sqrt{14} from -28.

$$x=\frac{-6\sqrt{14}-28}{-14}$$

Divide -28-6\sqrt{14} by -14.

$$x=\frac{3\sqrt{14}}{7}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2-\frac{3\sqrt{14}}{7} for x_{1} and 2+\frac{3\sqrt{14}}{7} for x_{2}.

$$-7x^{2}+28x-10=-7\left(x-\left(-\frac{3\sqrt{14}}{7}+2\right)\right)\left(x-\left(\frac{3\sqrt{14}}{7}+2\right)\right)$$

Evaluate

$$-7x^{2}+28x-10$$