Solve 3x^2-2-8x^2+6+5x

$$3 x ^ { 2 } - 2 - 8 x ^ { 2 } + 6 + 5 x$$

Evaluate

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

Solution Steps

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$-5x^{2}-2+6+5x$$

Add -2 and 6 to get 4.

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

Factor

$$-5\left(x-\left(-\frac{\sqrt{105}}{10}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{105}}{10}+\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$factor(-5x^{2}-2+6+5x)$$

Add -2 and 6 to get 4.

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

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.

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

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.

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

Square 5.

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

Multiply -4 times -5.

$$x=\frac{-5±\sqrt{25+20\times 4}}{2\left(-5\right)}$$

Multiply 20 times 4.

$$x=\frac{-5±\sqrt{25+80}}{2\left(-5\right)}$$

Add 25 to 80.

$$x=\frac{-5±\sqrt{105}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-5±\sqrt{105}}{-10}$$

Now solve the equation x=\frac{-5±\sqrt{105}}{-10} when ± is plus. Add -5 to \sqrt{105}.

$$x=\frac{\sqrt{105}-5}{-10}$$

Divide -5+\sqrt{105} by -10.

$$x=-\frac{\sqrt{105}}{10}+\frac{1}{2}$$

Now solve the equation x=\frac{-5±\sqrt{105}}{-10} when ± is minus. Subtract \sqrt{105} from -5.

$$x=\frac{-\sqrt{105}-5}{-10}$$

Divide -5-\sqrt{105} by -10.

$$x=\frac{\sqrt{105}}{10}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}-\frac{\sqrt{105}}{10} for x_{1} and \frac{1}{2}+\frac{\sqrt{105}}{10} for x_{2}.

$$-5x^{2}+5x+4=-5\left(x-\left(-\frac{\sqrt{105}}{10}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{105}}{10}+\frac{1}{2}\right)\right)$$