Solve (3x^2-x+5)+(-8x^2+3x-1)

$$(3 { x }^{ 2 } -x+5)+(-8 { x }^{ 2 } +3x-1)$$

Evaluate

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

Solution Steps

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

$$-5x^{2}-x+5+3x-1$$

Combine -x and 3x to get 2x.

$$-5x^{2}+2x+5-1$$

Subtract 1 from 5 to get 4.

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

Factor

$$-5\left(x-\frac{1-\sqrt{21}}{5}\right)\left(x-\frac{\sqrt{21}+1}{5}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-5x^{2}-x+5+3x-1)$$

Combine -x and 3x to get 2x.

$$factor(-5x^{2}+2x+5-1)$$

Subtract 1 from 5 to get 4.

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

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.

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

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.

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

Square 2.

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

Multiply -4 times -5.

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

Multiply 20 times 4.

$$x=\frac{-2±\sqrt{4+80}}{2\left(-5\right)}$$

Add 4 to 80.

$$x=\frac{-2±\sqrt{84}}{2\left(-5\right)}$$

Take the square root of 84.

$$x=\frac{-2±2\sqrt{21}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-2±2\sqrt{21}}{-10}$$

Now solve the equation x=\frac{-2±2\sqrt{21}}{-10} when ± is plus. Add -2 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}-2}{-10}$$

Divide -2+2\sqrt{21} by -10.

$$x=\frac{1-\sqrt{21}}{5}$$

Now solve the equation x=\frac{-2±2\sqrt{21}}{-10} when ± is minus. Subtract 2\sqrt{21} from -2.

$$x=\frac{-2\sqrt{21}-2}{-10}$$

Divide -2-2\sqrt{21} by -10.

$$x=\frac{\sqrt{21}+1}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{21}}{5} for x_{1} and \frac{1+\sqrt{21}}{5} for x_{2}.

$$-5x^{2}+2x+4=-5\left(x-\frac{1-\sqrt{21}}{5}\right)\left(x-\frac{\sqrt{21}+1}{5}\right)$$