Solve 10x^2+45x+14

$$10 { x }^{ 2 } +45x+14$$

Factor

$$10\left(x-\left(-\frac{\sqrt{1465}}{20}-\frac{9}{4}\right)\right)\left(x-\left(\frac{\sqrt{1465}}{20}-\frac{9}{4}\right)\right)$$

Steps Using the Quadratic Formula

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.

$$10x^{2}+45x+14=0$$

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.

$$x=\frac{-45±\sqrt{45^{2}-4\times 10\times 14}}{2\times 10}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\times 10\times 14}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-45±\sqrt{2025-40\times 14}}{2\times 10}$$

Multiply -40 times 14.

$$x=\frac{-45±\sqrt{2025-560}}{2\times 10}$$

Add 2025 to -560.

$$x=\frac{-45±\sqrt{1465}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-45±\sqrt{1465}}{20}$$

Now solve the equation x=\frac{-45±\sqrt{1465}}{20} when ± is plus. Add -45 to \sqrt{1465}.

$$x=\frac{\sqrt{1465}-45}{20}$$

Divide -45+\sqrt{1465} by 20.

$$x=\frac{\sqrt{1465}}{20}-\frac{9}{4}$$

Now solve the equation x=\frac{-45±\sqrt{1465}}{20} when ± is minus. Subtract \sqrt{1465} from -45.

$$x=\frac{-\sqrt{1465}-45}{20}$$

Divide -45-\sqrt{1465} by 20.

$$x=-\frac{\sqrt{1465}}{20}-\frac{9}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{9}{4}+\frac{\sqrt{1465}}{20} for x_{1} and -\frac{9}{4}-\frac{\sqrt{1465}}{20} for x_{2}.

$$10x^{2}+45x+14=10\left(x-\left(\frac{\sqrt{1465}}{20}-\frac{9}{4}\right)\right)\left(x-\left(-\frac{\sqrt{1465}}{20}-\frac{9}{4}\right)\right)$$

Evaluate

$$10x^{2}+45x+14$$