Problem 97 - Introductory Algebra for College Students

Question

Solve: \(x(2 x+9)=5\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -\frac{1}{2}\) and \(x = 5\).

Step 1: Expand the Left Side

First, you need to distribute \(x\) on the left side of the equation to get rid of the parenthesis. This leads to \(2x^2 + 9x = 5\).

Step 2: Bring All Terms to One Side

Move all terms to the left side of the equation. You do this by subtracting 5 from both sides, leading to \(2x^2 + 9x - 5 = 0\).

Step 3: Use Quadratic Equation

Now solve for \(x\) using the quadratic formula: \[x = -\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\] In this case, \(a=2\), \(b=9\), and \(c=-5\). Substituting the given values and simplifying will generate the solutions for \(x\).

Key Concepts

Distributing in Algebra

Distributing in algebra is a fundamental concept that involves eliminating parentheses by multiplying a single term across terms inside the parentheses. This step is often necessary when simplifying expressions or equations.
In the exercise, we are given the expression \[x(2x+9)\]which means we need to distribute \(x\) over \(2x+9\).
To distribute correctly:

• Multiply \(x\) by the first term inside the parentheses: \(x \times 2x = 2x^2\).
• Then, multiply \(x\) by the second term: \(x \times 9 = 9x\).

After distribution, the equation becomes \[2x^2 + 9x = 5\].
This step makes the equation easier to manage and sets the stage for the next phase of solving it.

Expanding Equations

Expanding equations means expressing the equation in a format where all algebraic terms are spread out and simplified. This process is vital in breaking down the equation into a solvable form, especially when dealing with quadratic equations.
Once you've distributed, the equation \[x(2x+9)=5\]becomes\[2x^2 + 9x = 5\].
Next, move all terms to one side of the equation to form a standard quadratic equation by subtracting 5 from both sides:\[2x^2 + 9x - 5 = 0\].

• The goal is to rewrite it in the standard quadratic form:\[ax^2 + bx + c = 0\], where it clearly shows the coefficients \(a\), \(b\), and \(c\).

Ensuring a precise expanded form allows you to accurately use algebraic tools like the quadratic formula for solving.

Solving Quadratic Equations

Once an equation is in the quadratic form\[2x^2 + 9x - 5 = 0\],the next step is solving it for \(x\). Quadratic equations can be solved by several methods, but here, we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
For our equation, identify the coefficients:

• \(a = 2\)
• \(b = 9\)
• \(c = -5\)

Plug these values into the quadratic formula:\[x = \frac{-9 \pm \sqrt{9^2 - 4 \times 2 \times (-5)}}{2 \times 2}\].
Simplify inside the square root and perform the arithmetic:

• First computation inside the square root is:\(9^2 - 4 \times 2 \times (-5)\) which simplifies to \(81 + 40 = 121\).
• Thus, \[x = \frac{-9 \pm \sqrt{121}}{4}=\frac{-9 \pm 11}{4}\].

Finally, solve for the two values of \(x\) by separating the plus and minus solutions:

• \(x = \frac{-9 + 11}{4} = \frac{2}{4} = \frac{1}{2}\)
• \(x = \frac{-9 - 11}{4} = \frac{-20}{4} = -5\)

These are the solutions for the quadratic equation.