Solve -2x^2-10x+5+3

$$- 2 x ^ { 2 } - 10 x + 5 + 3$$

Factor

$$-2\left(x-\frac{-\sqrt{41}-5}{2}\right)\left(x-\frac{\sqrt{41}-5}{2}\right)$$

Steps Using the Quadratic Formula

Add 5 and 3 to get 8.

$$factor(-2x^{2}-10x+8)$$

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.

$$-2x^{2}-10x+8=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-10\right)±\sqrt{100+8\times 8}}{2\left(-2\right)}$$

Multiply 8 times 8.

$$x=\frac{-\left(-10\right)±\sqrt{100+64}}{2\left(-2\right)}$$

Add 100 to 64.

$$x=\frac{-\left(-10\right)±\sqrt{164}}{2\left(-2\right)}$$

Take the square root of 164.

$$x=\frac{-\left(-10\right)±2\sqrt{41}}{2\left(-2\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{41}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{10±2\sqrt{41}}{-4}$$

Now solve the equation x=\frac{10±2\sqrt{41}}{-4} when ± is plus. Add 10 to 2\sqrt{41}.

$$x=\frac{2\sqrt{41}+10}{-4}$$

Divide 10+2\sqrt{41} by -4.

$$x=\frac{-\sqrt{41}-5}{2}$$

Now solve the equation x=\frac{10±2\sqrt{41}}{-4} when ± is minus. Subtract 2\sqrt{41} from 10.

$$x=\frac{10-2\sqrt{41}}{-4}$$

Divide 10-2\sqrt{41} by -4.

$$x=\frac{\sqrt{41}-5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5-\sqrt{41}}{2} for x_{1} and \frac{-5+\sqrt{41}}{2} for x_{2}.

$$-2x^{2}-10x+8=-2\left(x-\frac{-\sqrt{41}-5}{2}\right)\left(x-\frac{\sqrt{41}-5}{2}\right)$$

Evaluate

$$8-10x-2x^{2}$$

Solution Steps

Add 5 and 3 to get 8.

$$-2x^{2}-10x+8$$