Problem 49 - Beginning and Intermediate Algebra

Question

Solve each equation. $$\frac{1}{2}(m+1)^{2}=-\frac{3}{4} m(m+5)-\frac{5}{2}$$

Step-by-Step Solution

Answer

The solutions for the given equation are \(m = -3\) and \(m = -\frac{4}{5}\).

Step 1: Simplify the equation

First, let's expand and simplify the given equation: \(\frac{1}{2}(m+1)^{2}=-\frac{3}{4} m(m+5)-\frac{5}{2}\) Distribute the 1/2 to the square of (m+1): \(\frac{1}{2}(m^2 + 2m + 1) = -\frac{3}{4} m(m+5)-\frac{5}{2}\) Now, multiply and simplify further: \(\frac{1}{2}m^2 + m + \frac{1}{2} = -\frac{3}{4}m^2 - \frac{15}{4}m -\frac{5}{2}\)

Step 2: Move all terms to one side

We now want to rearrange the equation to be in the form \(ax^2 + bx + c = 0\), so we need to move all the terms to one side: \(\frac{1}{2}m^2 + m + \frac{1}{2} + \frac{3}{4}m^2 + \frac{15}{4}m +\frac{5}{2} = 0\)

Step 3: Combine like terms

Next, combine like terms (m^2, m, and constants) on the left-hand side: \(\frac{5}{4}m^2 + \frac{19}{4}m + 3 = 0\)

Step 4: Apply the quadratic formula

Now, we can apply the quadratic formula to find the value(s) of m: Let a = \(\frac{5}{4}\), b = \(\frac{19}{4}\), and c = 3. \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) \(m = \frac{-\frac{19}{4} \pm \sqrt{(\frac{19}{4})^2 - 4(\frac{5}{4})(3)}}{2(\frac{5}{4})}\)

Step 5: Simplify the expression under the square root

Simplify the expression under the square root: \((\frac{19}{4})^2 - 4(\frac{5}{4})(3) = \frac{361}{16} - \frac{60}{4} = \frac{361}{16} - \frac{240}{16} = \frac{121}{16}\)

Step 6: Insert the simplified expression back into the quadratic formula

Now, insert the simplified expression back into the quadratic formula: \(m = \frac{-\frac{19}{4} \pm \sqrt{\frac{121}{16}}}{2(\frac{5}{4})}\) \(m = \frac{-\frac{19}{4} \pm \frac{11}{4}}{\frac{5}{2}}\)

Step 7: Calculate the two possible values for m

Now, we'll calculate the two possible values for m: \(m_1 = \frac{-\frac{19}{4} - \frac{11}{4}}{\frac{5}{2}}\) \(m_1 = \frac{-\frac{30}{4}}{\frac{5}{2}}\) \(m_1 = \frac{-30}{10} = -3\) \(m_2 = \frac{-\frac{19}{4} + \frac{11}{4}}{\frac{5}{2}}\) \(m_2 = \frac{-\frac{8}{4}}{\frac{5}{2}}\) \(m_2 = \frac{-8}{10} = -\frac{4}{5}\) So, the solutions for the given equation are \(m = -3\) and \(m = -\frac{4}{5}\).

Key Concepts

Quadratic Formula

Quadratic equations are like puzzles that need to be solved using specific rules. These equations typically are in the format:

• \( ax^2 + bx + c = 0 \)

The quadratic formula provides a reliable method to find solutions for such equations. By substituting the values of \(a\), \(b\), and \(c\) from the quadratic equation, we use the quadratic formula:

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

This formula involves calculating the discriminant (\(b^2 - 4ac\)), which decides how many solutions a quadratic equation can have. If the discriminant is positive, we have two real solutions. If it's zero, there's one solution. And if negative, no real solutions exist. Understanding this formula is key to unlocking the mysteries of any quadratic equation you might face!

Solving Equations

Solving quadratic equations often involves transforming an equation to a standard form. This involves moving all terms to one side and setting the equation equal to zero, resulting in an equation like:

• \( ax^2 + bx + c = 0 \)

Every term in the equation serves a purpose: \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term. By rearranging and simplifying the equation, you can then apply methods like factoring, completing the square, or using the quadratic formula to find the solution. It's all about restructuring the problem into a form that is easier to solve!

Expand and Simplify Expressions

When you encounter an equation with expressions in parentheses or expressions that involve powers, such as \((m+1)^2\), the first step is to expand them. Expanding an expression means rewriting it without parentheses, by multiplying out the terms:- For \((m+1)^2\), it becomes \(m^2 + 2m + 1\) after expanding.Once expanded, each part of the equation can be simplified by performing the multiplications and divisions. Simplifying is crucial because it lays the groundwork for easier manipulation and solution of the equation later on. This step takes a complex expression and breaks it down into manageable parts.

Combining Like Terms

Once you have simplified the expressions, the next step is combining like terms. Like terms have the same variable and the same power, for example, combining \(m^2\) terms or \(m\) terms:

• \( \frac{1}{2}m^2 \) and \( \frac{3}{4}m^2 \) combine to give \( \frac{5}{4}m^2 \)
• \( m \) and \( \frac{15}{4}m \) combine to form \( \frac{19}{4}m \)

This process simplifies the equation further, making it easier to solve. By combining like terms, you're essentially cleaning up the equation, which helps in seeing patterns or solutions more clearly. It's like tidying up a messy room - once it's organized, you can find things much more easily!