Solve for B 9^2+B^2=13^2

Solution:

Step 1:

Compute the square of $9$. $81 + B^{2} = (13)^{2}$

Step 2:

Compute the square of $13$. $81 + B^{2} = 169$

Step 3:

Isolate the $B^2$ term by transferring all other terms to the opposite side of the equation.

Step 3.1:

Subtract $81$ from both sides. $B^{2} = 169 - 81$

Step 3.2:

Perform the subtraction. $B^{2} = 88$

Step 4:

Extract the square root of both sides to solve for $B$. $B = \pm \sqrt{88}$

Step 5:

Simplify the square root of $88$.

Step 5.1:

Express $88$ as a product of its prime factors.

Step 5.1.1:

Factor out $4$ from $88$. $B = \pm \sqrt{4 \cdot 22}$

Step 5.1.2:

Express $4$ as $2^2$. $B = \pm \sqrt{2^{2} \cdot 22}$

Step 5.2:

Remove terms from under the radical assuming they are perfect squares. $B = \pm 2\sqrt{22}$

Step 6:

Determine both the positive and negative solutions.

Step 6.1:

Apply the positive sign from $\pm$ to find the first solution. $B = 2\sqrt{22}$

Step 6.2:

Apply the negative sign from $\pm$ to find the second solution. $B = -2\sqrt{22}$

Step 6.3:

Combine both solutions. $B = 2\sqrt{22}, -2\sqrt{22}$

Step 7:

Present the solution in various forms.

Exact Form: $B = 2\sqrt{22}, -2\sqrt{22}$

Decimal Form: $B \approx 9.38083151, -9.38083151$

Knowledge Notes:

The problem at hand is a simple algebraic equation involving squares and square roots. Here are the relevant knowledge points:

1. Squaring a Number: This is the process of multiplying a number by itself. For example, $9^2 = 81$ and $13^2 = 169$.

2. Algebraic Manipulation: To solve for $B$, we need to isolate $B^2$ on one side of the equation. This involves moving terms across the equal sign, which requires us to perform the inverse operation (e.g., subtracting if we're moving an addition term).

3. Square Roots: Taking the square root is the inverse operation of squaring. For a positive number $A$, if $A = B^2$, then $B$ is either $\sqrt{A}$ or $-\sqrt{A}$.

4. Simplifying Square Roots: When a number under a square root can be factored into a perfect square multiplied by another number, the square root can be simplified by taking the square root of the perfect square out of the radical. For instance, $\sqrt{88}$ can be simplified because $88 = 4 \cdot 22$ and $4$ is a perfect square.

5. Positive and Negative Roots: When we take the square root of both sides of an equation, we must consider both the positive and negative square roots because squaring either a positive or negative number yields a positive result.

6. Exact vs. Decimal Form: Solutions to equations can be expressed exactly using square roots or approximately using decimals. Exact forms are more precise, while decimal forms are often used for practical measurements or when an approximate value is sufficient.