Solve (-2x+7x-9)+(14x^2-9x-14)

$$(-2x+7x-9)+(14 { x }^{ 2 } -9x-14)$$

Evaluate

$$14x^{2}-4x-23$$

Solution Steps

Combine -2x and 7x to get 5x.

$$5x-9+14x^{2}-9x-14$$

Combine 5x and -9x to get -4x.

$$-4x-9+14x^{2}-14$$

Subtract 14 from -9 to get -23.

$$-4x-23+14x^{2}$$

Factor

$$14\left(x-\left(-\frac{\sqrt{326}}{14}+\frac{1}{7}\right)\right)\left(x-\left(\frac{\sqrt{326}}{14}+\frac{1}{7}\right)\right)$$

Steps Using the Quadratic Formula

Combine -2x and 7x to get 5x.

$$factor(5x-9+14x^{2}-9x-14)$$

Combine 5x and -9x to get -4x.

$$factor(-4x-9+14x^{2}-14)$$

Subtract 14 from -9 to get -23.

$$factor(-4x-23+14x^{2})$$

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.

$$14x^{2}-4x-23=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 14\left(-23\right)}}{2\times 14}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 14\left(-23\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-\left(-4\right)±\sqrt{16-56\left(-23\right)}}{2\times 14}$$

Multiply -56 times -23.

$$x=\frac{-\left(-4\right)±\sqrt{16+1288}}{2\times 14}$$

Add 16 to 1288.

$$x=\frac{-\left(-4\right)±\sqrt{1304}}{2\times 14}$$

Take the square root of 1304.

$$x=\frac{-\left(-4\right)±2\sqrt{326}}{2\times 14}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{326}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{4±2\sqrt{326}}{28}$$

Now solve the equation x=\frac{4±2\sqrt{326}}{28} when ± is plus. Add 4 to 2\sqrt{326}.

$$x=\frac{2\sqrt{326}+4}{28}$$

Divide 4+2\sqrt{326} by 28.

$$x=\frac{\sqrt{326}}{14}+\frac{1}{7}$$

Now solve the equation x=\frac{4±2\sqrt{326}}{28} when ± is minus. Subtract 2\sqrt{326} from 4.

$$x=\frac{4-2\sqrt{326}}{28}$$

Divide 4-2\sqrt{326} by 28.

$$x=-\frac{\sqrt{326}}{14}+\frac{1}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{7}+\frac{\sqrt{326}}{14} for x_{1} and \frac{1}{7}-\frac{\sqrt{326}}{14} for x_{2}.

$$14x^{2}-4x-23=14\left(x-\left(\frac{\sqrt{326}}{14}+\frac{1}{7}\right)\right)\left(x-\left(-\frac{\sqrt{326}}{14}+\frac{1}{7}\right)\right)$$