Problem 17 - Elementary Algebra

Question

For the following problems, solve the equations. $$ s^{2}-9 s+8=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(s_1 = 8\) and \(s_2 = 1\).

Step 1: Identify the coefficients

Here, the coefficients for the given equation \(s^2 - 9s + 8 = 0\) are: \(a = 1\), \(b = -9\), and \(c = 8\).

Step 2: Plug the values into the quadratic formula

Now, plug the coefficients into the quadratic formula: $$ s = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(8)}}{2(1)} $$

Step 3: Simplify the expression

Next, simplify the equation step by step: $$ s = \frac{9 \pm \sqrt{81 - 32}}{2} $$ $$ s = \frac{9 \pm \sqrt{49}}{2} $$ $$ s = \frac{9 \pm 7}{2} $$

Step 4: Find the two solutions

Finally, solve for the two possible values of \(s\): $$ s_1 = \frac{9 + 7}{2} = \frac{16}{2} = 8 $$ $$ s_2 = \frac{9 - 7}{2} = \frac{2}{2} = 1 $$ So, the solutions for the given equation are \(s_1 = 8\) and \(s_2 = 1\).

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra, and it involves finding the values that make the equation true. In the context of quadratic equations, solving typically means finding the value or values of the variable that satisfy the equation.
In practice, we follow a series of steps to find these solutions:

• Identify the type of equation you are working with (e.g., linear, quadratic).
• For quadratic equations specifically, use the standard form: \( ax^2 + bx + c = 0 \).
• Determine the method to solve: Factoring, Quadratic Formula, or Completing the Square.

In the equation given, \( s^2 - 9s + 8 = 0 \), it’s a quadratic type because the highest power of \( s \) is 2. We use the quadratic formula since it provides a sure-shot way to find solutions when other methods like factoring might be cumbersome.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It allows you to find the roots (or solutions) even when factoring is difficult or impossible.
The quadratic formula is expressed as:
\[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula enables you to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \).
Let's break it down:

• \(-b\) changes the sign of \(b\).
• The \(\pm\) symbol indicates that there will be two solutions, one with plus and one with minus.
• \(\sqrt{b^2 - 4ac}\) is the discriminant, which determines the nature of the roots.

Using this formula in the original problem, the coefficients \(a = 1\), \(b = -9\), and \(c = 8\) lead to solutions \(s_1 = 8\) and \(s_2 = 1\) by simplifying step by step.

Algebraic Solutions

Algebraic solutions are the numerical or symbolic answers obtained through algebraic methods. For quadratic equations, this involves calculations based on the quadratic formula or other solving methods.
The critical part of reaching an algebraic solution is ensuring every step is followed accurately:

• Substitute the coefficients into the quadratic formula.
• Carry out each arithmetic operation precisely.
• Simplify carefully to avoid mistakes.

In our example, starting from:
\[ s = \frac{9 \pm \sqrt{49}}{2} \]The simplification gives us:

• \(s_1 = \frac{16}{2} = 8\), achieved by taking the plus sign.
• \(s_2 = \frac{2}{2} = 1\), achieved by taking the minus sign.

An algebraic solution gives precise answers, turning a problem into clear solutions.