Solve 4x^2+48x+45

$$4 { x }^{ 2 } +48x+45$$

Factor

$$4\left(x-\left(-\frac{3\sqrt{11}}{2}-6\right)\right)\left(x-\left(\frac{3\sqrt{11}}{2}-6\right)\right)$$

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.

$$4x^{2}+48x+45=0$$

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.

$$x=\frac{-48±\sqrt{48^{2}-4\times 4\times 45}}{2\times 4}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 4\times 45}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-48±\sqrt{2304-16\times 45}}{2\times 4}$$

Multiply -16 times 45.

$$x=\frac{-48±\sqrt{2304-720}}{2\times 4}$$

Add 2304 to -720.

$$x=\frac{-48±\sqrt{1584}}{2\times 4}$$

Take the square root of 1584.

$$x=\frac{-48±12\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-48±12\sqrt{11}}{8}$$

Now solve the equation x=\frac{-48±12\sqrt{11}}{8} when ± is plus. Add -48 to 12\sqrt{11}.

$$x=\frac{12\sqrt{11}-48}{8}$$

Divide -48+12\sqrt{11} by 8.

$$x=\frac{3\sqrt{11}}{2}-6$$

Now solve the equation x=\frac{-48±12\sqrt{11}}{8} when ± is minus. Subtract 12\sqrt{11} from -48.

$$x=\frac{-12\sqrt{11}-48}{8}$$

Divide -48-12\sqrt{11} by 8.

$$x=-\frac{3\sqrt{11}}{2}-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6+\frac{3\sqrt{11}}{2} for x_{1} and -6-\frac{3\sqrt{11}}{2} for x_{2}.

$$4x^{2}+48x+45=4\left(x-\left(\frac{3\sqrt{11}}{2}-6\right)\right)\left(x-\left(-\frac{3\sqrt{11}}{2}-6\right)\right)$$

Evaluate

$$4x^{2}+48x+45$$