Solve 1(x^2+5x-6)+(-6+x^2+1)

$$1 ( x ^ { 2 } + 5 x - 6 ) + ( - 6 + x ^ { 2 } + 1 )$$

Evaluate

$$2x^{2}+5x-11$$

Solution Steps

Use the distributive property to multiply 1 by x^{2}+5x-6.

$$x^{2}+5x-6-6+x^{2}+1$$

Subtract 6 from -6 to get -12.

$$x^{2}+5x-12+x^{2}+1$$

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

$$2x^{2}+5x-12+1$$

Add -12 and 1 to get -11.

$$2x^{2}+5x-11$$

Factor

$$2\left(x-\frac{-\sqrt{113}-5}{4}\right)\left(x-\frac{\sqrt{113}-5}{4}\right)$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1 by x^{2}+5x-6.

$$factor(x^{2}+5x-6-6+x^{2}+1)$$

Subtract 6 from -6 to get -12.

$$factor(x^{2}+5x-12+x^{2}+1)$$

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

$$factor(2x^{2}+5x-12+1)$$

Add -12 and 1 to get -11.

$$factor(2x^{2}+5x-11)$$

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.

$$2x^{2}+5x-11=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-11\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-11\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-11\right)}}{2\times 2}$$

Multiply -8 times -11.

$$x=\frac{-5±\sqrt{25+88}}{2\times 2}$$

Add 25 to 88.

$$x=\frac{-5±\sqrt{113}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±\sqrt{113}}{4}$$

Now solve the equation x=\frac{-5±\sqrt{113}}{4} when ± is plus. Add -5 to \sqrt{113}.

$$x=\frac{\sqrt{113}-5}{4}$$

Now solve the equation x=\frac{-5±\sqrt{113}}{4} when ± is minus. Subtract \sqrt{113} from -5.

$$x=\frac{-\sqrt{113}-5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{113}}{4} for x_{1} and \frac{-5-\sqrt{113}}{4} for x_{2}.

$$2x^{2}+5x-11=2\left(x-\frac{\sqrt{113}-5}{4}\right)\left(x-\frac{-\sqrt{113}-5}{4}\right)$$