Solving Quadratic Equations

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 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.