Problem 81 - Beginning and Intermediate Algebra

Question

Find the indicated values for the following polynomial functions. $$H(b)=b^{2}+3 .\( Find \)b\( so that \)H(b)=19$$

Step-by-Step Solution

Answer

There are two possible values for b: \(b = 4\) and \(b = -4\).

Step 1: Set up the equation

We need to set up the equation $$H(b) = 19$$, so we will replace \(H(b)\) in the given polynomial function with 19: $$19 = b^2 + 3$$

Step 2: Solve for b

Now, we will solve for b by subtracting 3 from both sides of the equation: $$19 - 3 = b^2$$ This simplifies to: $$16 = b^2$$

Step 3: Find the square root

To find the value of b, we take the square root of both sides of the equation: $$b = \pm \sqrt{16}$$

Step 4: Get the final answer

We find out that, $$b = \pm 4$$ So, there are two possible values for b: $$b = 4$$ and $$b = -4$$.

Key Concepts

Solving Equations

Solving equations is the process of finding the values of variables that make an equation true. When you see a function like \(H(b) = b^2 + 3\) set equal to a number, in this case 19, it means we need to find the value of \(b\) such that the equation holds true. To solve the equation \(H(b) = 19\), we can follow these steps:

• Write down the equation: \(19 = b^2 + 3\).
• Simplify by isolating \(b^2\) on one side. This is done by subtracting 3 from both sides: \(16 = b^2\).

Now you have a simpler equation, \(b^2 = 16\), which states that \(b\) squared equals 16.

Square Roots

Taking the square root is an operation that helps solve equations, especially quadratic ones like \(b^2 = 16\). When you take the square root of a number, you find the value which, when multiplied by itself, gives the original number. For \(b^2 = 16\):

• The square root of 16 is 4.
• Don't forget, square roots can be positive and negative. So, \(b = \pm 4\).

This means that \(b = 4\) and \(b = -4\) are both solutions because \((4)^2 = 16\) and \((-4)^2 = 16\), both satisfy the original equation.

Algebraic Expressions

Algebraic expressions require understanding of letters and numbers combined using operations like addition, subtraction, multiplication, and exponentiation. They express a value as evidenced by \(H(b) = b^2 + 3\).An algebraic expression can describe a quantity in more abstract terms:

• \(b^2\) means \(b\) is multiplied by itself, crucial for defining polynomial functions.
• The number 3 here is a constant, and acts as a shift in the function's output.

Breaking down expressions into simpler parts and understanding how to manipulate them is essential in solving and comprehending algebraic problems. You can use these principles to interpret and solve for values in more complicated expressions and equations.