Solve 225-x^2+34x

$$225- { x }^{ 2 } +34x$$

Factor

$$-\left(x-\left(17-\sqrt{514}\right)\right)\left(x-\left(\sqrt{514}+17\right)\right)$$

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.

$$-x^{2}+34x+225=0$$

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.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-1\right)\times 225}}{2\left(-1\right)}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-1\right)\times 225}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-34±\sqrt{1156+4\times 225}}{2\left(-1\right)}$$

Multiply 4 times 225.

$$x=\frac{-34±\sqrt{1156+900}}{2\left(-1\right)}$$

Add 1156 to 900.

$$x=\frac{-34±\sqrt{2056}}{2\left(-1\right)}$$

Take the square root of 2056.

$$x=\frac{-34±2\sqrt{514}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-34±2\sqrt{514}}{-2}$$

Now solve the equation x=\frac{-34±2\sqrt{514}}{-2} when ± is plus. Add -34 to 2\sqrt{514}.

$$x=\frac{2\sqrt{514}-34}{-2}$$

Divide -34+2\sqrt{514} by -2.

$$x=17-\sqrt{514}$$

Now solve the equation x=\frac{-34±2\sqrt{514}}{-2} when ± is minus. Subtract 2\sqrt{514} from -34.

$$x=\frac{-2\sqrt{514}-34}{-2}$$

Divide -34-2\sqrt{514} by -2.

$$x=\sqrt{514}+17$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17-\sqrt{514} for x_{1} and 17+\sqrt{514} for x_{2}.

$$-x^{2}+34x+225=-\left(x-\left(17-\sqrt{514}\right)\right)\left(x-\left(\sqrt{514}+17\right)\right)$$

Evaluate

$$225+34x-x^{2}$$