Solve -15x^2+4x-3x^2+2

$$- 15 x ^ { 2 } + 4 x - 3 x ^ { 2 } + 2$$

Factor

$$-18\left(x-\frac{1-\sqrt{10}}{9}\right)\left(x-\frac{\sqrt{10}+1}{9}\right)$$

Steps Using the Quadratic Formula

Combine -15x^{2} and -3x^{2} to get -18x^{2}.

$$factor(-18x^{2}+4x+2)$$

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.

$$-18x^{2}+4x+2=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-18\right)\times 2}}{2\left(-18\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-18\right)\times 2}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-4±\sqrt{16+72\times 2}}{2\left(-18\right)}$$

Multiply 72 times 2.

$$x=\frac{-4±\sqrt{16+144}}{2\left(-18\right)}$$

Add 16 to 144.

$$x=\frac{-4±\sqrt{160}}{2\left(-18\right)}$$

Take the square root of 160.

$$x=\frac{-4±4\sqrt{10}}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-4±4\sqrt{10}}{-36}$$

Now solve the equation x=\frac{-4±4\sqrt{10}}{-36} when ± is plus. Add -4 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}-4}{-36}$$

Divide -4+4\sqrt{10} by -36.

$$x=\frac{1-\sqrt{10}}{9}$$

Now solve the equation x=\frac{-4±4\sqrt{10}}{-36} when ± is minus. Subtract 4\sqrt{10} from -4.

$$x=\frac{-4\sqrt{10}-4}{-36}$$

Divide -4-4\sqrt{10} by -36.

$$x=\frac{\sqrt{10}+1}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{10}}{9} for x_{1} and \frac{1+\sqrt{10}}{9} for x_{2}.

$$-18x^{2}+4x+2=-18\left(x-\frac{1-\sqrt{10}}{9}\right)\left(x-\frac{\sqrt{10}+1}{9}\right)$$

Evaluate

$$2+4x-18x^{2}$$

Solution Steps

Combine -15x^{2} and -3x^{2} to get -18x^{2}.

$$-18x^{2}+4x+2$$