Problem 31 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$8 s^{2}+16 s=90$$

Step-by-Step Solution

Answer

The solutions are \(s = -\frac{9}{2}\) and \(s = \frac{5}{2}\).

Step 1: Move all terms to one side

The given equation is \(8s^2 + 16s = 90\). To set this equation to zero, subtract 90 from both sides: \(8s^2 + 16s - 90 = 0\). This is a standard form quadratic equation.

Step 2: Factor out the greatest common factor

In the expression \(8s^2 + 16s - 90\), notice that 2 is a common factor of all terms. Factor out 2 to obtain \(2(4s^2 + 8s - 45) = 0\).

Step 3: Factor the quadratic trinomial

Now, focus on factoring \(4s^2 + 8s - 45\). First, look for two numbers that multiply to \(4 \times -45 = -180\) and add to 8. These numbers are 18 and -10. Rewrite the middle term, 8, using these numbers: \(4s^2 + 18s - 10s - 45\).

Step 4: Group and factor by grouping

Group the terms in pairs: \((4s^2 + 18s) + (-10s - 45)\). Factor out the greatest common factor of each group: \(2s(2s + 9) - 5(2s + 9)\).

Step 5: Complete the factorization

Notice the common binomial factor \((2s + 9)\), factor it out: \((2s + 9)(2s - 5)\). Therefore, the complete factorization of the original quadratic equation is: \(2(2s + 9)(2s - 5) = 0\).

Step 6: Solve for the roots

Set each factor equal to zero: \(2s + 9 = 0\) or \(2s - 5 = 0\). Solve these equations: \(2s = -9\) leading to \(s = -\frac{9}{2}\), and \(2s = 5\) leading to \(s = \frac{5}{2}\).

Key Concepts

Factorization

Factorization is a method used to simplify and solve quadratic equations by expressing them as a product of simpler polynomials. In the simplest form, a quadratic equation can be written as \(ax^2 + bx + c = 0\). Our goal in factorization is to express the quadratic equation as \((px + q)(rx + s) = 0\), where \(p, q, r,\) and \(s\) are constants. This process involves finding numbers that multiply to give a specific value and add up to another.
For instance, in the exercise given, the equation \(8s^2 + 16s - 90 = 0\) had to be factored by first removing the greatest common factor from all terms, which was 2. This simplifies the equation to \(2(4s^2 + 8s - 45) = 0\). Next, we factor the trinomial \(4s^2 + 8s - 45\) by finding numbers that multiply to \(-180\) and sum to 8, precisely through trial and error. These numbers were 18 and -10.
Important steps in factorization include:

• Identifying and factoring out the greatest common factor
• Finding pairs of numbers that satisfy the product and sum conditions
• Grouping and factoring by grouping methods if needed

Mastering factorization allows one to break down complex quadratics into simple linear factors, making them easier to work with.

Roots of Equations

The roots of a quadratic equation are the solutions, or the values of the variable that satisfy the equation when it equals zero. Solving the equation \(ax^2 + bx + c = 0\) involves finding these roots. In our example, after factoring the original equation, we have \(2(2s + 9)(2s - 5) = 0\).
To find the roots, we solve for \(s\) in each factor separately by setting them to zero:

• \(2s + 9 = 0\), solving gives \(s = -\frac{9}{2}\)
• \(2s - 5 = 0\), solving gives \(s = \frac{5}{2}\)

These computations show that \(s = -\frac{9}{2}\) and \(s = \frac{5}{2}\) are the roots of the quadratic equation.
Roots are essential, not only to solve equations but also to understand the behavior of quadratic functions. They provide the points where the graph of the quadratic function intersects the x-axis.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that involves several strategies and methods to arrive at a solution, particularly with quadratic equations. This process often requires logical thinking and the ability to break down a problem into manageable parts.
Let's highlight the problem-solving process with our quadratic equation example:

• **Understanding the problem:** Begin by recognizing that solving a quadratic equation usually involves finding roots and restructuring the equation appropriately.
• **Strategizing**: We decide factorization is the best method here due to its efficiency in detecting the roots when dealing directly with an equation like \(8s^2 + 16s = 90\).
• **Executing the plan:** Moves include setting the equation to zero, factoring out common elements, factoring using methods like grouping, and then solving for each root.
• **Reviewing results:** Verify the calculated roots by substituting them back into the original equation to ensure they satisfy the equation.

Effective problem solving depends on a strong grasp of mathematical concepts and the ability to adapt methods based on the problem at hand. It is not just a procedure, but an essential skill to cultivate for all mathematical endeavors.