Solve 7+10x-5x^2

$$7+10x-5 { x }^{ 2 }$$

Factor

$$-5\left(x-\left(-\frac{2\sqrt{15}}{5}+1\right)\right)\left(x-\left(\frac{2\sqrt{15}}{5}+1\right)\right)$$

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.

$$-5x^{2}+10x+7=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\times 7}}{2\left(-5\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-5\right)\times 7}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-10±\sqrt{100+20\times 7}}{2\left(-5\right)}$$

Multiply 20 times 7.

$$x=\frac{-10±\sqrt{100+140}}{2\left(-5\right)}$$

Add 100 to 140.

$$x=\frac{-10±\sqrt{240}}{2\left(-5\right)}$$

Take the square root of 240.

$$x=\frac{-10±4\sqrt{15}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-10±4\sqrt{15}}{-10}$$

Now solve the equation x=\frac{-10±4\sqrt{15}}{-10} when ± is plus. Add -10 to 4\sqrt{15}.

$$x=\frac{4\sqrt{15}-10}{-10}$$

Divide -10+4\sqrt{15} by -10.

$$x=-\frac{2\sqrt{15}}{5}+1$$

Now solve the equation x=\frac{-10±4\sqrt{15}}{-10} when ± is minus. Subtract 4\sqrt{15} from -10.

$$x=\frac{-4\sqrt{15}-10}{-10}$$

Divide -10-4\sqrt{15} by -10.

$$x=\frac{2\sqrt{15}}{5}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{2\sqrt{15}}{5} for x_{1} and 1+\frac{2\sqrt{15}}{5} for x_{2}.

$$-5x^{2}+10x+7=-5\left(x-\left(-\frac{2\sqrt{15}}{5}+1\right)\right)\left(x-\left(\frac{2\sqrt{15}}{5}+1\right)\right)$$

Evaluate

$$7+10x-5x^{2}$$