Solve 4x^2+21x-36

$$4 { x }^{ 2 } +21x-36$$

Factor

$$4\left(x-\frac{-3\sqrt{113}-21}{8}\right)\left(x-\frac{3\sqrt{113}-21}{8}\right)$$

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.

$$4x^{2}+21x-36=0$$

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.

$$x=\frac{-21±\sqrt{21^{2}-4\times 4\left(-36\right)}}{2\times 4}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\times 4\left(-36\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-21±\sqrt{441-16\left(-36\right)}}{2\times 4}$$

Multiply -16 times -36.

$$x=\frac{-21±\sqrt{441+576}}{2\times 4}$$

Add 441 to 576.

$$x=\frac{-21±\sqrt{1017}}{2\times 4}$$

Take the square root of 1017.

$$x=\frac{-21±3\sqrt{113}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-21±3\sqrt{113}}{8}$$

Now solve the equation x=\frac{-21±3\sqrt{113}}{8} when ± is plus. Add -21 to 3\sqrt{113}.

$$x=\frac{3\sqrt{113}-21}{8}$$

Now solve the equation x=\frac{-21±3\sqrt{113}}{8} when ± is minus. Subtract 3\sqrt{113} from -21.

$$x=\frac{-3\sqrt{113}-21}{8}$$

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{113}}{8} for x_{1} and \frac{-21-3\sqrt{113}}{8} for x_{2}.

$$4x^{2}+21x-36=4\left(x-\frac{3\sqrt{113}-21}{8}\right)\left(x-\frac{-3\sqrt{113}-21}{8}\right)$$

Evaluate

$$4x^{2}+21x-36$$