Solve 2x^2-6x+3-4x-17x^2

$$2 { x }^{ 2 } -6x+3-4x-17 { x }^{ 2 }$$

Evaluate

$$3-10x-15x^{2}$$

Solution Steps

Combine -6x and -4x to get -10x.

$$2x^{2}-10x+3-17x^{2}$$

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

$$-15x^{2}-10x+3$$

Factor

$$-15\left(x-\left(-\frac{\sqrt{70}}{15}-\frac{1}{3}\right)\right)\left(x-\left(\frac{\sqrt{70}}{15}-\frac{1}{3}\right)\right)$$

Steps Using the Quadratic Formula

Combine -6x and -4x to get -10x.

$$factor(2x^{2}-10x+3-17x^{2})$$

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

$$factor(-15x^{2}-10x+3)$$

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.

$$-15x^{2}-10x+3=0$$

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.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-15\right)\times 3}}{2\left(-15\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-15\right)\times 3}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-\left(-10\right)±\sqrt{100+60\times 3}}{2\left(-15\right)}$$

Multiply 60 times 3.

$$x=\frac{-\left(-10\right)±\sqrt{100+180}}{2\left(-15\right)}$$

Add 100 to 180.

$$x=\frac{-\left(-10\right)±\sqrt{280}}{2\left(-15\right)}$$

Take the square root of 280.

$$x=\frac{-\left(-10\right)±2\sqrt{70}}{2\left(-15\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{70}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{10±2\sqrt{70}}{-30}$$

Now solve the equation x=\frac{10±2\sqrt{70}}{-30} when ± is plus. Add 10 to 2\sqrt{70}.

$$x=\frac{2\sqrt{70}+10}{-30}$$

Divide 10+2\sqrt{70} by -30.

$$x=-\frac{\sqrt{70}}{15}-\frac{1}{3}$$

Now solve the equation x=\frac{10±2\sqrt{70}}{-30} when ± is minus. Subtract 2\sqrt{70} from 10.

$$x=\frac{10-2\sqrt{70}}{-30}$$

Divide 10-2\sqrt{70} by -30.

$$x=\frac{\sqrt{70}}{15}-\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{3}-\frac{\sqrt{70}}{15} for x_{1} and -\frac{1}{3}+\frac{\sqrt{70}}{15} for x_{2}.

$$-15x^{2}-10x+3=-15\left(x-\left(-\frac{\sqrt{70}}{15}-\frac{1}{3}\right)\right)\left(x-\left(\frac{\sqrt{70}}{15}-\frac{1}{3}\right)\right)$$