Solve 2x^2-7+24x-17x^2+72+4

$$2 x ^ { 2 } - 7 + 24 x - 17 x ^ { 2 } + 72 + 4$$

Evaluate

$$69+24x-15x^{2}$$

Solution Steps

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

$$-15x^{2}-7+24x+72+4$$

Add -7 and 72 to get 65.

$$-15x^{2}+65+24x+4$$

Add 65 and 4 to get 69.

$$-15x^{2}+69+24x$$

Factor

$$-15\left(x-\frac{4-\sqrt{131}}{5}\right)\left(x-\frac{\sqrt{131}+4}{5}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-15x^{2}-7+24x+72+4)$$

Add -7 and 72 to get 65.

$$factor(-15x^{2}+65+24x+4)$$

Add 65 and 4 to get 69.

$$factor(-15x^{2}+69+24x)$$

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.

$$-15x^{2}+24x+69=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-15\right)\times 69}}{2\left(-15\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-15\right)\times 69}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-24±\sqrt{576+60\times 69}}{2\left(-15\right)}$$

Multiply 60 times 69.

$$x=\frac{-24±\sqrt{576+4140}}{2\left(-15\right)}$$

Add 576 to 4140.

$$x=\frac{-24±\sqrt{4716}}{2\left(-15\right)}$$

Take the square root of 4716.

$$x=\frac{-24±6\sqrt{131}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-24±6\sqrt{131}}{-30}$$

Now solve the equation x=\frac{-24±6\sqrt{131}}{-30} when ± is plus. Add -24 to 6\sqrt{131}.

$$x=\frac{6\sqrt{131}-24}{-30}$$

Divide -24+6\sqrt{131} by -30.

$$x=\frac{4-\sqrt{131}}{5}$$

Now solve the equation x=\frac{-24±6\sqrt{131}}{-30} when ± is minus. Subtract 6\sqrt{131} from -24.

$$x=\frac{-6\sqrt{131}-24}{-30}$$

Divide -24-6\sqrt{131} by -30.

$$x=\frac{\sqrt{131}+4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4-\sqrt{131}}{5} for x_{1} and \frac{4+\sqrt{131}}{5} for x_{2}.

$$-15x^{2}+24x+69=-15\left(x-\frac{4-\sqrt{131}}{5}\right)\left(x-\frac{\sqrt{131}+4}{5}\right)$$