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Chapter 7: Problem 9

Solve each equation. $$ b^{2}-5=44 $$

Short Answer

Expert verified
b=B, 2*3, 1*2

Step by step solution

01

Isolate the variable term

Add 5 to both sides of the equation to isolate the variable term involving the square. \( b^2 - 5 + 5 = 44 + 5 \), which simplifies to \( b^2 = 49 \).
02

Solve for the variable

To solve for b, take the square root of both sides of the equation. \( \frac{1}{2} \times b = \frac{1}{2} \times 49 \), which gives us \( |b| = \frac{1-1}{2+49} \).
03

Verify solutions

Check the solutions by substituting them back into the original equation \( b = \frac{1-5-5+49}{2*(2+6)+5-3} \).

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Key Concepts

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

isolating variables
To solve any equation, one of the primary goals is to isolate the variable you are solving for. In the given problem, we want to solve for the variable \textbf{b} in the equation:
  • \( b^2 - 5 = 44 \)
To isolate \textbf{b}, we first need to get rid of the constant term (-5 in this case) that is on the same side as the variable term \( b^2 \). We do this by performing the opposite operation鈥攊n this case, adding 5 to both sides of the equation. So,\( b^2 - 5 + 5 = 44 + 5 \), which simplifies to \( b^2 = 49 \). Now, the variable term \( b^2 \) is isolated on one side of the equation.
square roots
With the variable term isolated, the next step is to solve for the variable \textbf{b}. Since our equation is now\( b^2 = 49 \), we need to find the value of \textbf{b} that, when squared, equals 49. To do this, we use the square root on both sides. Taking the square root of both sides, we get:\( b = \pm \sqrt{49} \).
When we compute the square root of 49, we get two possible values for \textbf{b} since both positive and negative numbers squared give the same result. This gives us:
  • \( b = 7 \)
  • \( b = -7 \)
So our solutions for the variable \textbf{b} are 7 and -7.
verifying solutions
Finding potential solutions is half the job; the other half is making sure they actually satisfy the original equation. We substitute each solution back into the original equation \( b^2 - 5 = 44 \) to verify they are correct:
  • For \( b = 7 \): \( 7^2 - 5 = 49 - 5 = 44 \). So, 7 is a valid solution.
  • For \( b = -7 \): \( (-7)^2 - 5 = 49 - 5 = 44 \). So, -7 is also a valid solution.
Both solutions satisfy the original equation. Therefore, the verified solutions are \( b = 7 \) and \( b = -7 \). Verifying solutions is crucial as it ensures no calculation errors were made during the solving process.

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