Solve x^2+x-56<0

$$x ^ { 2 } + x - 56 < 0$$

Solve for x

$$x\in \left(-8,7\right)$$

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.

$$x^{2}+x-56=0$$

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 -56 for c in the quadratic formula.

$$x=\frac{-1±\sqrt{1^{2}-4\times 1\left(-56\right)}}{2}$$

Do the calculations.

$$x=\frac{-1±15}{2}$$

Solve the equation x=\frac{-1±15}{2} when ± is plus and when ± is minus.

$$x=7 x=-8$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-7\right)\left(x+8\right)<0$$

For the product to be negative, x-7 and x+8 have to be of the opposite signs. Consider the case when x-7 is positive and x+8 is negative.

$$x-7>0 x+8<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+8 is positive and x-7 is negative.

$$x+8>0 x-7<0$$

The solution satisfying both inequalities is x\in \left(-8,7\right).

$$x\in \left(-8,7\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-8,7\right)$$