Solve 4x^2-21x-72

$$4 { x }^{ 2 } -21x-72$$

Factor

$$4\left(x-\frac{21-3\sqrt{177}}{8}\right)\left(x-\frac{3\sqrt{177}+21}{8}\right)$$

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.

$$4x^{2}-21x-72=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\left(-72\right)}}{2\times 4}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 4\left(-72\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-16\left(-72\right)}}{2\times 4}$$

Multiply -16 times -72.

$$x=\frac{-\left(-21\right)±\sqrt{441+1152}}{2\times 4}$$

Add 441 to 1152.

$$x=\frac{-\left(-21\right)±\sqrt{1593}}{2\times 4}$$

Take the square root of 1593.

$$x=\frac{-\left(-21\right)±3\sqrt{177}}{2\times 4}$$

The opposite of -21 is 21.

$$x=\frac{21±3\sqrt{177}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{21±3\sqrt{177}}{8}$$

Now solve the equation x=\frac{21±3\sqrt{177}}{8} when ± is plus. Add 21 to 3\sqrt{177}.

$$x=\frac{3\sqrt{177}+21}{8}$$

Now solve the equation x=\frac{21±3\sqrt{177}}{8} when ± is minus. Subtract 3\sqrt{177} from 21.

$$x=\frac{21-3\sqrt{177}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21+3\sqrt{177}}{8} for x_{1} and \frac{21-3\sqrt{177}}{8} for x_{2}.

$$4x^{2}-21x-72=4\left(x-\frac{3\sqrt{177}+21}{8}\right)\left(x-\frac{21-3\sqrt{177}}{8}\right)$$

Evaluate

$$4x^{2}-21x-72$$