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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 form
a_{n}x^{n} + ··· + a_{2}x^{2} + a_{1}x + a_{0}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:
ax^{2} + bx + c = 0According 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:
$${x}^{2}+\frac{b}{a}x=-\frac{c}{a}$$We can convert the left side of this equation to a perfect square by adding b^{2}/4a^{2}, which is equal to (b/2a)^{2}:
$\text{Leftside:}{x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}={\left(x+\frac{b}{2a}\right)}^{2}$Having added a value to the left side, we must now add that same value, b^{2} ⁄ 4a^{2}, to the right side:
$${\left(x+\frac{b}{2a}\right)}^{2}=-\frac{c}{a}+\frac{{b}^{2}}{4{a}^{2}}$$The common denominator on the right side is 4a^{2}. Rearranging the right side, we obtain the following:
$${\left(x+\frac{b}{2a}\right)}^{2}=\frac{{b}^{2}-4ac}{4{a}^{2}}$$Taking the square root of both sides and solving for x,
$$x+\frac{b}{2a}=\frac{\pm \sqrt{{b}^{2}-4ac}}{2a}$$ $$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$$This equation, known as the quadratic formula, has two roots:
$$x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a}\text{and}x=\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}$$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.
Solution: