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If 2ba2=18, what is the value of b? (1) a2=1,156 (2) a>0

Short Answer

Expert verified
The value of b is 587.

Step by step solution

01

Substitute given value of a2

Given that a2=1156, substitute 1156 for a2 in the equation 2ba2=18. This gives: 2b1156=18
02

Solve for 2b

To isolate 2b, add 1156 to both sides of the equation: 2b=18+1156 2b=1174
03

Divide by 2 to find b

To find b, divide both sides of the equation by 2: b=11742 b=587
04

Verify positivity condition

The second condition states that a>0. Since a2=1156, a can be either 34 or 34. To satisfy a>0, we choose a=34. However, b does not change as it only depends on a2. Thus, b=587 holds.

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Key Concepts

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

quadratic equations
In algebra, a quadratic equation is a second-order polynomial equation in a single variable. The general form is ax2+bx+c=0,where a, b, and c are constants, and x is the variable. Quadratic equations often appear in a variety of contexts, and solving them can be crucial for many mathematical tasks. For instance, in the given exercise, the equation we started with is not in the standard form, but can be easily related to quadratic forms through algebraic manipulations. It's important to recognize when an equation is quadratic, as it informs how we may approach solving it.
variable isolation
Variable isolation involves manipulating an equation to get the variable of interest alone on one side. This is often required to solve for that variable.

In our exercise, the goal is to find the value of b. We start with the equation: 2ba2=18 To isolate 2b, we add a2 to both sides:2b=18+a2 Once isolated, the next steps become more straightforward. This process is crucial in algebraic problem-solving and is especially common in both linear and nonlinear equations.
substitution method
The substitution method is a technique to solve equations or systems of equations. This involves replacing a variable with a given value or expression.

In our case, we are given a2=1156. By substituting 1156 for a2 in the equation 2ba2=18, it simplifies to: 2b1156=18. This simplification makes it easier to solve for b since it reduces the original equation to an equation in a single variable.
algebraic manipulations
Algebraic manipulations involve operations like addition, subtraction, multiplication, and division to simplify or solve equations.

In the provided solution, we performed several key manipulations:
  • Substituting 1156 for a2: 2b1156=18
  • Adding 1156 to both sides to isolate 2b: 2b=1174
  • Dividing both sides by 2 to solve for b: b=11742=587
These steps demonstrate how algebraic manipulations can simplify and ultimately solve equations, showcasing their importance in mathematical problem-solving.

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