Solve 8x-94-x^2-3x+3x^2+27

$$8 x - 94 - x ^ { 2 } - 3 x + 3 x ^ { 2 } + 27$$

Evaluate

$$2x^{2}+5x-67$$

Solution Steps

Combine 8x and -3x to get 5x.

$$5x-94-x^{2}+3x^{2}+27$$

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

$$5x-94+2x^{2}+27$$

Add -94 and 27 to get -67.

$$5x-67+2x^{2}$$

Factor

$$2\left(x-\frac{-\sqrt{561}-5}{4}\right)\left(x-\frac{\sqrt{561}-5}{4}\right)$$

Steps Using the Quadratic Formula

Combine 8x and -3x to get 5x.

$$factor(5x-94-x^{2}+3x^{2}+27)$$

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

$$factor(5x-94+2x^{2}+27)$$

Add -94 and 27 to get -67.

$$factor(5x-67+2x^{2})$$

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.

$$2x^{2}+5x-67=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-67\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-67\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-67\right)}}{2\times 2}$$

Multiply -8 times -67.

$$x=\frac{-5±\sqrt{25+536}}{2\times 2}$$

Add 25 to 536.

$$x=\frac{-5±\sqrt{561}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±\sqrt{561}}{4}$$

Now solve the equation x=\frac{-5±\sqrt{561}}{4} when ± is plus. Add -5 to \sqrt{561}.

$$x=\frac{\sqrt{561}-5}{4}$$

Now solve the equation x=\frac{-5±\sqrt{561}}{4} when ± is minus. Subtract \sqrt{561} from -5.

$$x=\frac{-\sqrt{561}-5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{561}}{4} for x_{1} and \frac{-5-\sqrt{561}}{4} for x_{2}.

$$2x^{2}+5x-67=2\left(x-\frac{\sqrt{561}-5}{4}\right)\left(x-\frac{-\sqrt{561}-5}{4}\right)$$