Solve x^2-x-42>0

$$x ^ { 2 } - x - 42 > 0$$

Solve for x

$$x\in \left(-\infty,-6\right)\cup \left(7,\infty\right)$$

Steps Using the Quadratic Formula

To solve the inequality, factor the left hand side. 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.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -1 for b, and -42 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\left(-42\right)}}{2}$$

Do the calculations.

$$x=\frac{1±13}{2}$$

Solve the equation x=\frac{1±13}{2} when ± is plus and when ± is minus.

$$x=7 x=-6$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-7\right)\left(x+6\right)>0$$

For the product to be positive, x-7 and x+6 have to be both negative or both positive. Consider the case when x-7 and x+6 are both negative.

$$x-7<0 x+6<0$$

The solution satisfying both inequalities is x<-6.

$$x<-6$$

Consider the case when x-7 and x+6 are both positive.

$$x+6>0 x-7>0$$

The solution satisfying both inequalities is x>7.

$$x>7$$

The final solution is the union of the obtained solutions.

$$x<-6\text{; }x>7$$