Solve (2x+1*3x+1)-x^2

$$(2x+1 \times 3x+1)- { x }^{ 2 }$$

Evaluate

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

Solution Steps

Multiply 1 and 3 to get 3.

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

Combine 2x and 3x to get 5x.

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

Factor

$$-\left(x-\frac{5-\sqrt{29}}{2}\right)\left(x-\frac{\sqrt{29}+5}{2}\right)$$

Steps Using the Quadratic Formula

Multiply 1 and 3 to get 3.

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

Combine 2x and 3x to get 5x.

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

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.

$$-x^{2}+5x+1=0$$

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.

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

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4}}{2\left(-1\right)}$$

Add 25 to 4.

$$x=\frac{-5±\sqrt{29}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±\sqrt{29}}{-2}$$

Now solve the equation x=\frac{-5±\sqrt{29}}{-2} when ± is plus. Add -5 to \sqrt{29}.

$$x=\frac{\sqrt{29}-5}{-2}$$

Divide -5+\sqrt{29} by -2.

$$x=\frac{5-\sqrt{29}}{2}$$

Now solve the equation x=\frac{-5±\sqrt{29}}{-2} when ± is minus. Subtract \sqrt{29} from -5.

$$x=\frac{-\sqrt{29}-5}{-2}$$

Divide -5-\sqrt{29} by -2.

$$x=\frac{\sqrt{29}+5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{29}}{2} for x_{1} and \frac{5+\sqrt{29}}{2} for x_{2}.

$$-x^{2}+5x+1=-\left(x-\frac{5-\sqrt{29}}{2}\right)\left(x-\frac{\sqrt{29}+5}{2}\right)$$