Mathematical functions of this form are known as second-order polynomials or, more commonly, quadratic functions. $$ax^2+bx+c=0$$

The solution or roots for any quadratic equation can be calculated using the following formula: $$x=\frac{−b±\sqrt{b^2−4ac}}{2a}$$

### Solving Quadratic Equations Example

Solve the quadratic equation $ 3x^2 + 13x − 10 = 0$ .

**Solution**

Solution Substituting the values a = 3, b = 13, c = −10 in the formula, we obtain: $$x=\frac{−13±\sqrt{(13)^2−4×3×(−10)}}{2×3}$$ $$x=\frac{−13±\sqrt{169+120}}{6}=\frac{−13±\sqrt{289}}{6}=\frac{−13±17}{6}$$

The two roots are therefore $$x=\frac{−13+17}{6}=\mathbf{\frac{2}{3}}\quad and\quad x=\frac{−13−17}{6}=\mathbf{−5}$$

As you can see in the example above, the mathematical solution for a quadratic equation can produce negative (or sometimes imaginary) roots. When solving quadratic equations that relate to scientific measurements, remember:

**Roots that you carry forward must be REAL**. At least in the realm that we operate in for introductory chemistry, imaginary numbers do not correspond to solutions you can use in calculations or measurements.

**Roots that you carry forward are (usually) POSITIVE .** We cannot have a negative value for moles or concentration, for example! The exception to this is if you have set up an ICE table “backwards” (with the reaction proceeding in the reverse direction) — it is possible to have a negative value for “

*x*” (i.e. the

*change*in moles or concentration can be negative) so long as none of your actual values of moles or concentration are negative.