Solve 2x^2-6-17x^2+11+4x

$$2 x ^ { 2 } - 6 - 17 x ^ { 2 } + 11 + 4 x$$

Evaluate

$$5+4x-15x^{2}$$

Solution Steps

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

$$-15x^{2}-6+11+4x$$

Add -6 and 11 to get 5.

$$-15x^{2}+5+4x$$

Factor

$$-15\left(x-\frac{2-\sqrt{79}}{15}\right)\left(x-\frac{\sqrt{79}+2}{15}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-15x^{2}-6+11+4x)$$

Add -6 and 11 to get 5.

$$factor(-15x^{2}+5+4x)$$

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.

$$-15x^{2}+4x+5=0$$

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.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-15\right)\times 5}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-4±\sqrt{16+60\times 5}}{2\left(-15\right)}$$

Multiply 60 times 5.

$$x=\frac{-4±\sqrt{16+300}}{2\left(-15\right)}$$

Add 16 to 300.

$$x=\frac{-4±\sqrt{316}}{2\left(-15\right)}$$

Take the square root of 316.

$$x=\frac{-4±2\sqrt{79}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-4±2\sqrt{79}}{-30}$$

Now solve the equation x=\frac{-4±2\sqrt{79}}{-30} when ± is plus. Add -4 to 2\sqrt{79}.

$$x=\frac{2\sqrt{79}-4}{-30}$$

Divide -4+2\sqrt{79} by -30.

$$x=\frac{2-\sqrt{79}}{15}$$

Now solve the equation x=\frac{-4±2\sqrt{79}}{-30} when ± is minus. Subtract 2\sqrt{79} from -4.

$$x=\frac{-2\sqrt{79}-4}{-30}$$

Divide -4-2\sqrt{79} by -30.

$$x=\frac{\sqrt{79}+2}{15}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2-\sqrt{79}}{15} for x_{1} and \frac{2+\sqrt{79}}{15} for x_{2}.

$$-15x^{2}+4x+5=-15\left(x-\frac{2-\sqrt{79}}{15}\right)\left(x-\frac{\sqrt{79}+2}{15}\right)$$