The Quadratic Formula: Videos & Practice Problems

The quadratic formula, expressed as $x=-b\pm \sqrt{b²-4ac}/2a$, is essential for solving equations of the form $\mathrm{ax}²+\mathrm{bx}+c=0$. It is particularly useful in chemical equilibrium problems, where ICE charts help determine equilibrium concentrations. Typically, only one of the two potential solutions is significant, often the positive value, which will be discussed further in the context of chemical equilibrium.

Quadratic Formula

The quadratic formula is a powerful tool used to solve algebraic equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are numerical coefficients, and \( x \) represents the variable we want to solve for. The formula is expressed as:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is particularly useful in various applications, including chemical equilibrium problems, where it helps determine equilibrium concentrations using ICE (Initial, Change, Equilibrium) charts. In these scenarios, the quadratic formula can yield two potential solutions for \( x \) due to the plus or minus sign. However, typically only one of these solutions is relevant, often the positive value, depending on the context of the problem.

To illustrate the application of the quadratic formula, consider the equation derived from a specific problem:

\( 4x^2 + 2.13 \times 10^{-4}x - 1.75 \times 10^{-5} = 0 \)

In this equation, \( a = 4 \), \( b = 2.13 \times 10^{-4} \), and \( c = -1.75 \times 10^{-5} \). Plugging these values into the quadratic formula, we first calculate the discriminant \( b^2 - 4ac \) to ensure it is non-negative, which is necessary for real solutions. The calculation proceeds as follows:

\( b^2 - 4ac = (2.13 \times 10^{-4})^2 - 4(4)(-1.75 \times 10^{-5}) \)

After determining the value of the discriminant, we can find the square root and substitute back into the quadratic formula. This will yield two potential solutions for \( x \): one from the addition and one from the subtraction of the square root term. For example, if the calculations yield:

\( x_1 = \frac{-2.13 \times 10^{-4} + 0.016735}{8} \approx 0.002065 \)

\( x_2 = \frac{-2.13 \times 10^{-4} - 0.016735}{8} \approx -0.002119 \)

In this case, \( x_1 \) is the viable solution, while \( x_2 \) may be discarded based on the context of the problem, particularly in chemical equilibrium scenarios where negative concentrations are not physically meaningful.

Understanding the quadratic formula and its application in solving equations is crucial, especially in fields like chemistry, where it aids in analyzing equilibrium states and concentrations effectively.

Here’s what students ask on this topic:

What is the quadratic formula and how is it used?

The quadratic formula is used to solve quadratic equations of the form $a^{2}x+bx+c=0$. The formula is expressed as $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. It is particularly useful in finding the roots of the equation, which are the values of x that satisfy the equation. In chemical equilibrium problems, it helps determine equilibrium concentrations using ICE charts.

When do you use the quadratic formula in chemical equilibrium problems?

The quadratic formula is used in chemical equilibrium problems when you need to solve for equilibrium concentrations. These problems often involve setting up an ICE chart (Initial, Change, Equilibrium) to organize the concentrations of reactants and products. When the equilibrium expression results in a quadratic equation of the form $a^{2}x+bx+c=0$, the quadratic formula can be applied to find the values of x. Typically, only one of the two possible solutions is physically meaningful, often the positive value.

How do you determine which solution to use from the quadratic formula?

When using the quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, you get two potential solutions for x. In the context of chemical equilibrium, only one solution is typically significant. This is usually the positive value, as negative concentrations are not physically meaningful. The context of the problem and the physical constraints will guide you in selecting the appropriate solution.

What are ICE charts and how do they relate to the quadratic formula?

ICE charts (Initial, Change, Equilibrium) are used in chemical equilibrium problems to organize and calculate the concentrations of reactants and products at equilibrium. They help set up the equilibrium expression, which can sometimes result in a quadratic equation of the form $a^{2}x+bx+c=0$. The quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, is then used to solve for the equilibrium concentrations.

Can the quadratic formula have more than one solution?

Yes, the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ provides two potential solutions for x due to the ± symbol. These solutions are the roots of the quadratic equation $a^{2}x+bx+c=0$. However, in practical applications like chemical equilibrium, only one solution is often significant, usually the positive value, as negative concentrations are not physically meaningful.