Questions: 2 b^2 + 100 = 0 b = □

Transcript text: \[ \begin{array}{l} 2 b^{2}+100=0 \\ b=\square \end{array} \]

Solution

Solution Steps

To solve the equation \(2b^2 + 100 = 0\), we need to isolate \(b\). First, subtract 100 from both sides to get \(2b^2 = -100\). Then, divide both sides by 2 to get \(b^2 = -50\). Finally, take the square root of both sides, keeping in mind that the square root of a negative number involves imaginary numbers.

Step 1: Isolate \( b^2 \)

Starting with the equation: \[ 2b^2 + 100 = 0 \] Subtract 100 from both sides: \[ 2b^2 = -100 \]

Step 2: Solve for \( b^2 \)

Divide both sides by 2: \[ b^2 = -50 \]

Step 3: Take the Square Root

Take the square root of both sides, remembering that the square root of a negative number involves imaginary numbers: \[ b = \pm \sqrt{-50} \] \[ b = \pm \sqrt{50}i \]

Step 4: Simplify the Square Root

Simplify \(\sqrt{50}\): \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \] Thus: \[ b = \pm 5\sqrt{2}i \]

Step 5: Approximate the Value

Approximating \(\sqrt{2}\) to four significant digits: \[ \sqrt{2} \approx 1.414 \] So: \[ b \approx \pm 5 \times 1.414i = \pm 7.071i \]