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Chapter 7: Problem 57

Solve each equation. $$(q+3)^{2}-(2 q-5)^{2}=0$$

Short Answer

Expert verified
The solutions to the equation \((q+3)^2-(2q-5)^2=0\) are \(q=8\) and \(q=\frac{2}{3}\).

Step by step solution

01

Rewrite the given equation using the difference of squares formula.

We can rewrite the equation using the difference of squares formula, which is: \[a^2 - b^2 = (a - b)(a + b).\] Apply this formula to the given equation: \[((q + 3) - (2q - 5))((q + 3) + (2q - 5)) = 0.\]
02

Simplify and expand the equation.

Now, we simplify and expand the equation: \[(q + 3 - 2q + 5)(q + 3 + 2q - 5) = 0,\] \[(-q + 8)(3q - 2) = 0.\]
03

Find the solutions for q.

Since the product of the two factors is 0, either one or both of the factors must be equal to 0. So, we have two possible cases: 1) \(-q + 8 = 0\) 2) \(3q -2 = 0\) Now we solve both cases: (1) \(-q + 8 = 0\), \(-q = -8\), \(q = 8\). (2) \(3q-2=0 \), \(3q = 2\), \(q = \frac{2}{3}\). Therefore, the solutions are \(q=8\) and \(q=\frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The concept of 'Difference of Squares' can be a great tool when faced with quadratic equations that follow a specific format. It essentially means that you have an equation in the form \(a^2 - b^2\). This form can be simplified to \((a - b)(a + b)\). For the given equation, this pattern becomes apparent: we have two squared terms subtracting each other. This allows us to use the difference of squares formula. We let \(a = q + 3\) and \(b = 2q - 5\), which helps us rewrite the equation in terms of these two factors. Once this step is clarified, the complexity of the original equation is reduced significantly to a simpler product of two brackets.
Equation Simplification
In mathematics, simplifying an equation is essential to find the solution easily. Once we've expressed our quadratic equation with the difference of squares formula, our next goal is to simplify these expressions. Consider the expression \((q + 3) - (2q - 5)\) and \((q + 3) + (2q - 5)\). By distributing and combining like terms, they simplify into \(-q + 8\) and \(3q - 2\) respectively. Keep in mind the order of operations: subtraction, addition, and then factor aggregation. Breaking down these brackets helps to visualize the problem in simpler terms. Making it less daunting to solve.
Quadratic Solutions
Once your equation has been simplified to a product like \((-q + 8)(3q - 2) = 0\), finding solutions becomes straightforward. The principle here is that if a product of two terms equals zero, then at least one of the terms must be zero. This leads us to set \(-q + 8 = 0\) and \(3q - 2 = 0\), and solve each equation separately. Solving the first gives \(q = 8\) by isolating \(q\). For the second equation, rearranging terms leads to \(q = \frac{2}{3}\). Collectively, these solutions show that the original equation has two possible values for \(q\), illustrating the typical outcome when dealing with quadratic equations.

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