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sessions for the long-time client package, will she be able to create this package for her clients? A) No, because the closest package that she can offer consists of three hot yoga and three zero gravity yoga sessions. B) No, because the closest package that she can offer consists of four hot yoga and four zero gravity yoga sessions. C) Yes, because she can offer five hot yoga and five zero gravity yoga sessions. D) Yes, because she can offer six hot yoga and six zero gravity yoga sessions. 13\. Cuthbert is conducting a chemistry experiment that calls for a number of chemicals to be mixed in various quantities. The one amount of which he is unsure is grams of potassium, \(p\). If Cuthbert is certain that \(\left(3 p^2+\right.\) \(14 p+24)-2\left(p^2+7 p+20\right)=0\), what is one possible value of \(3 p+6\), the exact number of grams of potassium that Cuthbert would like to use for this experiment? A) 20 B) 18 C) 12 D) 10

Step by step solution

01

Simplify the given equation

We are given the equation: \((3p^2 + 14p + 24) - 2(p^2 + 7p + 20) = 0\) First, distribute the -2 into the second term inside the parentheses: \(3p^2 + 14p + 24 - 2p^2 - 14p - 40 = 0\) Now, combine like terms: \(3p^2 - 2p^2 + 14p - 14p + 24 - 40 = 0\) \(p^2 - 16 = 0\)

02

Solve the quadratic equation

We have the simplified equation: \(p^2 - 16 = 0\) Now we can see it's a difference of squares, so we can factor it as: \((p + 4)(p - 4) = 0\) So, by the zero-product property, p = -4 or p = 4

03

Calculate 3p + 6 for each value of p

1) For p = -4: 3p + 6 = 3(-4) + 6 = -12 + 6 = -6. 2) For p = 4: 3p + 6 = 3(4) + 6 = 12 + 6 = 18 As we found two possible values of 3p+6, we need to check the answer choices. Looking at the options, we can see that -6 is not present, but 18 is. The correct answer is: B) 18

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Key Concepts

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

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They represent polynomial equations of degree two, typically written in the form \( ax^2 + bx + c = 0 \). In this equation, \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero (otherwise, it wouldn't be a quadratic).

These equations are essential because they describe parabolas in a coordinate system. Understanding how to work with them allows us to solve a wide range of mathematical problems, from physics applications to everyday decision-making scenarios.

To find the solution or roots of a quadratic equation, various techniques can be employed, such as factoring, completing the square, or using the quadratic formula. In our example, the equation \( p^2 - 16 = 0 \) is solved using factoring, which brings us to how these approaches are interconnected.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators that represent mathematical relations or rules. They are the building blocks for forming equations and are crucial for expressing real-world phenomena in mathematical terms.

• Variables: Symbols like \(p\) or \(x\) that stand for unknown values.
• Coefficients: Numbers that multiply variables in terms like \(3p^2\).
• Constants: Fixed values like \(24\) in our example.
• Operators: Symbols indicating mathematical operations, e.g., \(+\), \(-\).

Understanding algebraic expressions is vital as they form the basis for creating equations and inequalities, allowing us to model and solve problems effectively.

For instance, simplifying the expression \((3p^2 + 14p + 24) - 2(p^2 + 7p + 20)\) involves distributing and then combining like terms to identify a solvable equation for \(p\). This process of simplification transforms a complex situation into a manageable equation.

Zero-Product Property

The zero-product property is a key principle in algebra used to solve quadratic equations. It states that if a product of two factors is zero, then at least one of the factors must be zero. Mathematically, if \(a \cdot b = 0\), then either \(a = 0\), \(b = 0\), or both.

This property provides a simple solution path once an equation is factored, as seen with \((p + 4)(p - 4) = 0\). Applying the zero-product property here means setting each factor to zero, leading to potential solutions, \(p = 4\) or \(p = -4\).

It's an efficient tool in problem solving, especially for students tackling SAT Math Problems, as it turns a quadratic equation into two simpler linear equations. Mastering this concept can greatly enhance the speed and ease of solving quadratic-related problems.

Factoring Techniques

Factoring is a critical method for solving quadratic equations, especially when they are presented in a factorable form like the difference of squares or trinomials. In factoring, the quadratic is rewritten as a product of binomials.

The equation \( p^2 - 16 = 0 \) is solved as a difference of squares, where \(a^2 - b^2 = (a + b)(a - b)\). This method quickly identifies \( (p + 4)(p - 4) = 0 \) as the factored form.

• Difference of Squares: Easily factor expressions like \( p^2 - 16 \) into \( (p+4)(p-4) \).
• Trinomials: Expressions like \(x^2 + bx + c\) that can sometimes be factored into two binomials.

Knowing when and how to apply factoring techniques is invaluable for swiftly solving equations and is a staple in preparing for SAT Math Problems. As these skills develop, tackling more complex equations becomes straightforward, enhancing math problem solving abilities for students.