Question

Solve the following equation: bb−5−2b+5=50b2−25
and show your work.

Answer 1

To solve the equation \( b^2 - 5 - 2b + 5 = 50b^2 - 25 \), we can simplify and rearrange the terms.

1. Combine like terms on the left side:

\[
b^2 - 2b - 5 + 5 = b^2 - 2b
\]

So the equation simplifies to:

\[
b^2 - 2b = 50b^2 - 25
\]

2. Now, bring all terms to one side of the equation:

\[
b^2 - 2b - 50b^2 + 25 = 0
\]

Combine the \( b^2 \) terms:

\[
-49b^2 - 2b + 25 = 0
\]

Multiplying the entire equation by -1 to make coefficients positive:

\[
49b^2 + 2b - 25 = 0
\]

3. Now, we can use the quadratic formula to solve for \( b \):

The quadratic formula is:

\[
b = \frac{{-B \pm \sqrt{{B^2 - 4AC}}}}{2A}
\]

For our equation \( 49b^2 + 2b - 25 = 0 \):
- \( A = 49 \)
- \( B = 2 \)
- \( C = -25 \)

4. Calculate \( B^2 - 4AC \):

\[
B^2 - 4AC = 2^2 - 4 \cdot 49 \cdot (-25) = 4 + 4900 = 4904
\]

5. Now substitute into the quadratic formula:

\[
b = \frac{{-2 \pm \sqrt{4904}}}{2 \cdot 49}
\]

6. Simplifying further, the square root of 4904:

\[
\sqrt{4904} = 2\sqrt{1226}
\]

Therefore, the equation becomes:

\[
b = \frac{{-2 \pm 2\sqrt{1226}}}{98}
\]

Simplifying this gives:

\[
b = \frac{{-1 \pm \sqrt{1226}}}{49}
\]

7. Finding the approximate values for \( b \):

First, calculate \( \sqrt{1226} \approx 34.95 \).

So we have two possible solutions:

\[
b_1 \approx \frac{{-1 + 34.95}}{49} = \frac{33.95}{49} \approx 0.692
\]

\[
b_2 \approx \frac{{-1 - 34.95}}{49} = \frac{-35.95}{49} \approx -0.734
\]

8. The approximate solutions for \( b \) are:

\[
b \approx 0.692 \quad \text{or} \quad b \approx -0.734
\]

Thus, the solutions to the equation \( b^2 - 5 - 2b + 5 = 50b^2 - 25 \) are approximately \( b \approx 0.692 \) and \( b \approx -0.734 \).