Solve 3x+14x^2+6x+1+9x

$$3 x + 14 x ^ { 2 } + 6 x + 1 + 9 x$$

Evaluate

$$14x^{2}+18x+1$$

Solution Steps

Combine 3x and 6x to get 9x.

$$9x+14x^{2}+1+9x$$

Combine 9x and 9x to get 18x.

$$18x+14x^{2}+1$$

Factor

$$14\left(x-\frac{-\sqrt{67}-9}{14}\right)\left(x-\frac{\sqrt{67}-9}{14}\right)$$

Steps Using the Quadratic Formula

Combine 3x and 6x to get 9x.

$$factor(9x+14x^{2}+1+9x)$$

Combine 9x and 9x to get 18x.

$$factor(18x+14x^{2}+1)$$

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.

$$14x^{2}+18x+1=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\times 14}}{2\times 14}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 14}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-18±\sqrt{324-56}}{2\times 14}$$

Add 324 to -56.

$$x=\frac{-18±\sqrt{268}}{2\times 14}$$

Take the square root of 268.

$$x=\frac{-18±2\sqrt{67}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-18±2\sqrt{67}}{28}$$

Now solve the equation x=\frac{-18±2\sqrt{67}}{28} when ± is plus. Add -18 to 2\sqrt{67}.

$$x=\frac{2\sqrt{67}-18}{28}$$

Divide -18+2\sqrt{67} by 28.

$$x=\frac{\sqrt{67}-9}{14}$$

Now solve the equation x=\frac{-18±2\sqrt{67}}{28} when ± is minus. Subtract 2\sqrt{67} from -18.

$$x=\frac{-2\sqrt{67}-18}{28}$$

Divide -18-2\sqrt{67} by 28.

$$x=\frac{-\sqrt{67}-9}{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+\sqrt{67}}{14} for x_{1} and \frac{-9-\sqrt{67}}{14} for x_{2}.

$$14x^{2}+18x+1=14\left(x-\frac{\sqrt{67}-9}{14}\right)\left(x-\frac{-\sqrt{67}-9}{14}\right)$$