Question

If \(2 b-a^{2}=18,\) what is the value of \(b ?\) (1) \(a^{2}=1,156\) (2) \(a>0\)

Short answer

The value of \(b\) is 587.

Step-by-step solution

Substitute given value of \(a^2\)

Given that \( a^2 = 1156 \), substitute \( 1156 \) for \( a^2 \) in the equation \( 2b - a^2 = 18 \). This gives: \[ 2b - 1156 = 18 \]

Solve for \(2b\)

To isolate \(2b\), add \(1156\) to both sides of the equation: \[ 2b = 18 + 1156 \] \[ 2b = 1174 \]

Divide by 2 to find \(b\)

To find \(b\), divide both sides of the equation by 2: \[ b = \frac{1174}{2} \] \[ b = 587 \]

Verify positivity condition

The second condition states that \( a > 0 \). Since \( a^2 = 1156 \), \( a \) can be either \( 34 \) or \( -34 \). To satisfy \( a > 0 \), we choose \( a = 34 \). However, \(b\) does not change as it only depends on \( a^2 \). Thus, \( b = 587 \) holds.

Key concepts

quadratic equations

In algebra, a quadratic equation is a second-order polynomial equation in a single variable. The general form is \[ ax^2 + bx + c = 0 \],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: \[ 2b - a^2 = 18 \] To isolate \( 2b \), we add \( a^2 \) to both sides:\[ 2b = 18 + a^2 \] 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 \( a^2 = 1156 \). By substituting \( 1156 \) for \( a^2 \) in the equation \[ 2b - a^2 = 18 \], it simplifies to: \[ 2b - 1156 = 18 \]. 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 \( a^2 \): \[ 2b - 1156 = 18 \]
• Adding \( 1156 \) to both sides to isolate \( 2b \): \[ 2b = 1174 \]
• Dividing both sides by 2 to solve for \( b \): \[ b = \frac{1174}{2} = 587 \]

These steps demonstrate how algebraic manipulations can simplify and ultimately solve equations, showcasing their importance in mathematical problem-solving.