Previous Essential Skills sections introduced many of the mathematical operations you need to solve chemical problems. We now introduce the quadratic formula, a mathematical relationship involving sums of powers in a single variable that you will need to apply to solve some of the problems in this chapter.
Mathematical expressions that involve a sum of powers in one or more variables (e.g., x) multiplied by coefficients (such as a) are called polynomials. Polynomials of a single variable have the general formanxn + ··· + a2x2 + a1x + a0
The highest power to which the variable in a polynomial is raised is called its order. Thus the polynomial shown here is of the nth order. For example, if n were 3, the polynomial would be third order.
A quadratic equation is a second-order polynomial equation in a single variable x:ax2 + bx + c = 0
According to the fundamental theorem of algebra, a second-order polynomial equation has two solutions—called roots—that can be found using a method called completing the square. In this method, we solve for x by first adding −c to both sides of the quadratic equation and then divide both sides by a:
We can convert the left side of this equation to a perfect square by adding b2/4a2, which is equal to (b/2a)2:
Having added a value to the left side, we must now add that same value, b2 ⁄ 4a2, to the right side:
The common denominator on the right side is 4a2. Rearranging the right side, we obtain the following:
Taking the square root of both sides and solving for x,
This equation, known as the quadratic formula, has two roots:
Thus we can obtain the solutions to a quadratic equation by substituting the values of the coefficients (a, b, c) into the quadratic formula.
When you apply the quadratic formula to obtain solutions to a quadratic equation, it is important to remember that one of the two solutions may not make sense or neither may make sense. There may be times, for example, when a negative solution is not reasonable or when both solutions require that a square root be taken of a negative number. In such cases, we simply discard any solution that is unreasonable and only report a solution that is reasonable. Skill Builder ES1 gives you practice using the quadratic formula.
Use the quadratic formula to solve for x in each equation. Report your answers to three significant figures.