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Magazine Chapter 9: Problem 77
Solve and round off the solutions to the nearest hundredth. $$ (2 x+1)(x+3)-(x+7)=(x+3) 2 $$
Short Answer
Expert verified
The solutions are approximately 1.61 and -3.11.
Step by step solution
01
Expand Both Sides
First, expand both sides of the equation. Start with the left side: \[ (2x+1)(x+3) - (x+7) = 2x(x+3) + 1(x+3) - x - 7 \]Now expand:\[ = 2x^2 + 6x + x + 3 - x - 7 \] Combine like terms:\[ = 2x^2 + 6x + 3 - 7 \] Simplify:\[ = 2x^2 + 5x - 4 \]Now for the right side:\[ (x+3)2 = 2(x+3) = 2x + 6 \] So the equation becomes:\[ 2x^2 + 5x - 4 = 2x + 6 \]
02
Rearrange the Equation
Move all terms to one side of the equation to set it to zero:\[ 2x^2 + 5x - 4 - 2x - 6 = 0 \] Combine the terms:\[ 2x^2 + 3x - 10 = 0 \]
03
Apply the Quadratic Formula
We now apply the quadratic formula to solve for \(x\):\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our equation, \(a = 2\), \(b = 3\), and \(c = -10\). Substitute these into the formula:\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-10)}}{2 \cdot 2} \] Simplify under the square root:\[ x = \frac{-3 \pm \sqrt{9 + 80}}{4} \] \[ = \frac{-3 \pm \sqrt{89}}{4} \]
04
Compute the Solutions
Calculate the solutions by evaluating the expression separately for both the plus and minus results from the quadratic formula:1. \(+\sqrt{89}: \) \[ x_1 = \frac{-3 + \sqrt{89}}{4} \approx \frac{-3 + 9.43}{4} \approx \frac{6.43}{4} \approx 1.61 \]2. \(-\sqrt{89}: \) \[ x_2 = \frac{-3 - \sqrt{89}}{4} \approx \frac{-3 - 9.43}{4} \approx \frac{-12.43}{4} \approx -3.11 \]
05
Round to the Nearest Hundredth
The solutions calculated earlier were already rounded to the nearest hundredth. Thus, we have:\[ x_1 \approx 1.61 \]\[ x_2 \approx -3.11 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the roots or solutions of the equation. Here's how it works:
- Identify \( a \), \( b \), and \( c \): These come from the standard form of the quadratic equation.
- Calculate the discriminant: \( b^2 - 4ac \). The value under the square root symbol.
- Determine the solutions: Use the '+' and '-' signs to find two potential solutions.
- Solve: Evaluate each solution by plugging values back into the equation.
Algebraic Expansion
Algebraic expansion involves multiplying terms to remove parentheses in an expression. Consider the expression \((2x + 1)(x + 3)\). To expand it:
- Distribute each term: Multiply each term in the first parenthesis by each term in the second. For instance, \(2x \times x \), \(2x \times 3 \), \(1 \times x \), \(1 \times 3 \).
- Write out the results: Combine them all to form: \(2x^2 + 6x + x + 3\).
- Simplify: Combine like terms which gives us \(2x^2 + 7x + 3\).
Combining Like Terms
Combining like terms is a method used to simplify algebraic expressions. Terms are 'like' if they share the same variable to the same power. Consider the expression we expanded: \(2x^2 + 6x + x + 3\). To simplify it:
- Identify like terms: Notice that \(6x\) and \(x\) are like terms since both have the variable \(x\).
- Combine the coefficients: Add the numbers in front of the like terms, which yields \(6 + 1 = 7\), thus \(7x\).
- Rewrite the expression: \(2x^2 + 7x + 3\).
Rounding Numbers
Rounding numbers is the process of adjusting values to a certain degree of precision, often to make them easier to work with. When asked to round a number to the nearest hundredth, you look at the digits after the decimal point.
- Locate the hundredth place: This is the second digit to the right of the decimal point.
- Check the digit after it: For instance, in the number 1.612, the hundredth place is '1'.
- Decide how to round: If the next digit is 5 or more, round up. So, 1.612 becomes 1.61.
- Adjust accordingly: Conversely, if the next digit is 4 or less, keep the number as is.
