Solve 2x^2-7-17x^2+9+5x

$$2 x ^ { 2 } - 7 - 17 x ^ { 2 } + 9 + 5 x$$

Evaluate

$$2+5x-15x^{2}$$

Solution Steps

Combine 2x^{2} and -17x^{2} to get -15x^{2}.

$$-15x^{2}-7+9+5x$$

Add -7 and 9 to get 2.

$$-15x^{2}+2+5x$$

Factor

$$-15\left(x-\left(-\frac{\sqrt{145}}{30}+\frac{1}{6}\right)\right)\left(x-\left(\frac{\sqrt{145}}{30}+\frac{1}{6}\right)\right)$$

Steps Using the Quadratic Formula

Combine 2x^{2} and -17x^{2} to get -15x^{2}.

$$factor(-15x^{2}-7+9+5x)$$

Add -7 and 9 to get 2.

$$factor(-15x^{2}+2+5x)$$

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.

$$-15x^{2}+5x+2=0$$

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.

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

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-15\right)\times 2}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-5±\sqrt{25+60\times 2}}{2\left(-15\right)}$$

Multiply 60 times 2.

$$x=\frac{-5±\sqrt{25+120}}{2\left(-15\right)}$$

Add 25 to 120.

$$x=\frac{-5±\sqrt{145}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-5±\sqrt{145}}{-30}$$

Now solve the equation x=\frac{-5±\sqrt{145}}{-30} when ± is plus. Add -5 to \sqrt{145}.

$$x=\frac{\sqrt{145}-5}{-30}$$

Divide -5+\sqrt{145} by -30.

$$x=-\frac{\sqrt{145}}{30}+\frac{1}{6}$$

Now solve the equation x=\frac{-5±\sqrt{145}}{-30} when ± is minus. Subtract \sqrt{145} from -5.

$$x=\frac{-\sqrt{145}-5}{-30}$$

Divide -5-\sqrt{145} by -30.

$$x=\frac{\sqrt{145}}{30}+\frac{1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{6}-\frac{\sqrt{145}}{30} for x_{1} and \frac{1}{6}+\frac{\sqrt{145}}{30} for x_{2}.

$$-15x^{2}+5x+2=-15\left(x-\left(-\frac{\sqrt{145}}{30}+\frac{1}{6}\right)\right)\left(x-\left(\frac{\sqrt{145}}{30}+\frac{1}{6}\right)\right)$$