Solve 2x^2-27-14x+3x^2

$$2 x ^ { 2 } - 27 - 14 x + 3 x ^ { 2 }$$

Factor

$$5\left(x-\frac{7-2\sqrt{46}}{5}\right)\left(x-\frac{2\sqrt{46}+7}{5}\right)$$

Steps Using the Quadratic Formula

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$factor(5x^{2}-27-14x)$$

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.

$$5x^{2}-14x-27=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\left(-27\right)}}{2\times 5}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 5\left(-27\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-14\right)±\sqrt{196-20\left(-27\right)}}{2\times 5}$$

Multiply -20 times -27.

$$x=\frac{-\left(-14\right)±\sqrt{196+540}}{2\times 5}$$

Add 196 to 540.

$$x=\frac{-\left(-14\right)±\sqrt{736}}{2\times 5}$$

Take the square root of 736.

$$x=\frac{-\left(-14\right)±4\sqrt{46}}{2\times 5}$$

The opposite of -14 is 14.

$$x=\frac{14±4\sqrt{46}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{14±4\sqrt{46}}{10}$$

Now solve the equation x=\frac{14±4\sqrt{46}}{10} when ± is plus. Add 14 to 4\sqrt{46}.

$$x=\frac{4\sqrt{46}+14}{10}$$

Divide 14+4\sqrt{46} by 10.

$$x=\frac{2\sqrt{46}+7}{5}$$

Now solve the equation x=\frac{14±4\sqrt{46}}{10} when ± is minus. Subtract 4\sqrt{46} from 14.

$$x=\frac{14-4\sqrt{46}}{10}$$

Divide 14-4\sqrt{46} by 10.

$$x=\frac{7-2\sqrt{46}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+2\sqrt{46}}{5} for x_{1} and \frac{7-2\sqrt{46}}{5} for x_{2}.

$$5x^{2}-14x-27=5\left(x-\frac{2\sqrt{46}+7}{5}\right)\left(x-\frac{7-2\sqrt{46}}{5}\right)$$

Evaluate

$$5x^{2}-14x-27$$

Solution Steps

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}-27-14x$$