Solve 9x^2-42x-3

$$9 { x }^{ 2 } -42x-3$$

Factor

$$9\left(x-\frac{7-2\sqrt{13}}{3}\right)\left(x-\frac{2\sqrt{13}+7}{3}\right)$$

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.

$$9x^{2}-42x-3=0$$

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.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 9\left(-3\right)}}{2\times 9}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\times 9\left(-3\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-42\right)±\sqrt{1764-36\left(-3\right)}}{2\times 9}$$

Multiply -36 times -3.

$$x=\frac{-\left(-42\right)±\sqrt{1764+108}}{2\times 9}$$

Add 1764 to 108.

$$x=\frac{-\left(-42\right)±\sqrt{1872}}{2\times 9}$$

Take the square root of 1872.

$$x=\frac{-\left(-42\right)±12\sqrt{13}}{2\times 9}$$

The opposite of -42 is 42.

$$x=\frac{42±12\sqrt{13}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{42±12\sqrt{13}}{18}$$

Now solve the equation x=\frac{42±12\sqrt{13}}{18} when ± is plus. Add 42 to 12\sqrt{13}.

$$x=\frac{12\sqrt{13}+42}{18}$$

Divide 42+12\sqrt{13} by 18.

$$x=\frac{2\sqrt{13}+7}{3}$$

Now solve the equation x=\frac{42±12\sqrt{13}}{18} when ± is minus. Subtract 12\sqrt{13} from 42.

$$x=\frac{42-12\sqrt{13}}{18}$$

Divide 42-12\sqrt{13} by 18.

$$x=\frac{7-2\sqrt{13}}{3}$$

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{13}}{3} for x_{1} and \frac{7-2\sqrt{13}}{3} for x_{2}.

$$9x^{2}-42x-3=9\left(x-\frac{2\sqrt{13}+7}{3}\right)\left(x-\frac{7-2\sqrt{13}}{3}\right)$$

Evaluate

$$9x^{2}-42x-3$$