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Magazine Chapter 8: Problem 34
Solve. See Examples 1 through \(5 .\). $$ 2(4 m-3)^{2}-9(4 m-3)=5 $$
Short Answer
Expert verified
The solutions for \( m \) are \( m = 2 \) and \( m = \frac{5}{8} \).
Step by step solution
01
Substitute Variable
Let \( u = 4m - 3 \) to simplify the equation. The equation becomes \( 2u^2 - 9u = 5 \).
02
Rearrange the Equation
Rearrange the equation into standard quadratic form: \( 2u^2 - 9u - 5 = 0 \).
03
Apply the Quadratic Formula
The quadratic formula is \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation, \( a = 2 \), \( b = -9 \), and \( c = -5 \). Substitute these values into the formula.
04
Calculate the Discriminant
Calculate the discriminant \( b^2 - 4ac \), which is \((-9)^2 - 4(2)(-5) = 81 + 40 = 121 \).
05
Solve for \( u \) using the Quadratic Formula
Since the discriminant is 121, it is a perfect square. Hence, \( u = \frac{9 \pm \sqrt{121}}{4} \). Simplify this to find \( u = \frac{9 + 11}{4} = 5 \) or \( u = \frac{9 - 11}{4} = -\frac{1}{2} \).
06
Back Substitute for \( m \)
Recall \( u = 4m - 3 \). When \( u = 5 \), \( 4m - 3 = 5 \). Solving gives \( 4m = 8 \), so \( m = 2 \). Similarly, when \( u = -\frac{1}{2} \), solve \( 4m - 3 = -\frac{1}{2} \).
07
Solve the Second Equation for \( m \)
From \( 4m - 3 = -\frac{1}{2} \), rearrange to \( 4m = \frac{5}{2} \). Therefore, \( m = \frac{5}{8} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a great way to simplify complex equations. It involves replacing part of the equation with a single variable, making the equation easier to manage and solve.
In the given problem, we set \( u = 4m - 3 \), turning a complicated expression into a simpler variable. This helps us focus on solving a standard quadratic equation.
Once we solve for \( u \), we can reverse the substitution to find the original variable, in this case, \( m \).
In the given problem, we set \( u = 4m - 3 \), turning a complicated expression into a simpler variable. This helps us focus on solving a standard quadratic equation.
Once we solve for \( u \), we can reverse the substitution to find the original variable, in this case, \( m \).
- Start by identifying a repeated or complex expression.
- Replace it with a simpler variable.
- Work through the simplified equation.
- Substitute back to find the original variable's value.
Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to find the roots of the equation. Here's the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the solutions \( x \) directly by substituting the coefficients \( a \), \( b \), and \( c \) from the equation.
In this exercise, once the substitution method is applied, we rearranged the equation into its standard form and identified \( a = 2 \), \( b = -9 \), and \( c = -5 \). After substituting these values into the quadratic formula, we were able to solve for \( u \).
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the solutions \( x \) directly by substituting the coefficients \( a \), \( b \), and \( c \) from the equation.
In this exercise, once the substitution method is applied, we rearranged the equation into its standard form and identified \( a = 2 \), \( b = -9 \), and \( c = -5 \). After substituting these values into the quadratic formula, we were able to solve for \( u \).
- Substitute coefficients into the formula.
- Calculate the value of the discriminant inside the formula.
- Use arithmetic operations to solve for the roots of the equation.
Discriminant
The discriminant of a quadratic equation provides key information about the nature of the equation's roots. It is part of the quadratic formula, represented as \( b^2 - 4ac \). Here鈥檚 what it tells you:
- If the discriminant is positive, there are two distinct real roots.- If it's zero, there is exactly one real root (a repeated root).- If it鈥檚 negative, there are no real roots, only complex ones.
In the problem, the discriminant is calculated as \((-9)^2 - 4(2)(-5) = 121\). Because 121 is positive and a perfect square, it indicates that there are two distinct and rational solutions for \( u \).
- If the discriminant is positive, there are two distinct real roots.- If it's zero, there is exactly one real root (a repeated root).- If it鈥檚 negative, there are no real roots, only complex ones.
In the problem, the discriminant is calculated as \((-9)^2 - 4(2)(-5) = 121\). Because 121 is positive and a perfect square, it indicates that there are two distinct and rational solutions for \( u \).
- Calculate \( b^2 - 4ac \) to find the discriminant.
- Determine root nature based on its value.
- A positive and perfect square result gives rational roots.
Perfect Square
Understanding the concept of a perfect square is important when solving quadratic equations, especially when using the quadratic formula. A perfect square is a number that is the square of an integer.
In the context of quadratic equations, finding that the discriminant is a perfect square, like 121 in our example, means that the solutions will be rational.
The roots \( \sqrt{121} = 11 \) help simplify the solution process since they can be expressed as exact rational numbers rather than decimals or complex numbers.
In the context of quadratic equations, finding that the discriminant is a perfect square, like 121 in our example, means that the solutions will be rational.
The roots \( \sqrt{121} = 11 \) help simplify the solution process since they can be expressed as exact rational numbers rather than decimals or complex numbers.
- A perfect square significantly simplifies solving quadratic equations.
- Identifying perfect squares helps in checking factorability.
- Rational results imply simpler interpretation and ease in substitution.
