Questions: ([Delta c][Delta c])/([0.159-Delta c])=4.60 cdot 10^-4

Solution Steps

The given equation is:

\[ \frac{[\Delta c][\Delta c]}{[0.159-\Delta c]}=4.60 \cdot 10^{-4} \]

This can be rewritten as:

\[ \frac{(\Delta c)^2}{0.159 - \Delta c} = 4.60 \times 10^{-4} \]

To solve for \(\Delta c\), we first multiply both sides by \(0.159 - \Delta c\) to eliminate the fraction:

\[ (\Delta c)^2 = 4.60 \times 10^{-4} \times (0.159 - \Delta c) \]

Expand the right side of the equation:

\[ (\Delta c)^2 = 4.60 \times 10^{-4} \times 0.159 - 4.60 \times 10^{-4} \times \Delta c \]

Simplify the expression:

\[ (\Delta c)^2 = 7.314 \times 10^{-5} - 4.60 \times 10^{-4} \Delta c \]

Rearrange the equation to form a quadratic equation:

\[ (\Delta c)^2 + 4.60 \times 10^{-4} \Delta c - 7.314 \times 10^{-5} = 0 \]

This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where:

• \(a = 1\)
• \(b = 4.60 \times 10^{-4}\)
• \(c = -7.314 \times 10^{-5}\)

Use the quadratic formula:

\[ \Delta c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute the values:

\[ \Delta c = \frac{-4.60 \times 10^{-4} \pm \sqrt{(4.60 \times 10^{-4})^2 - 4 \times 1 \times (-7.314 \times 10^{-5})}}{2 \times 1} \]

Calculate the discriminant:

\[ (4.60 \times 10^{-4})^2 = 2.116 \times 10^{-7} \] \[ 4 \times 1 \times (-7.314 \times 10^{-5}) = -2.9256 \times 10^{-4} \] \[ b^2 - 4ac = 2.116 \times 10^{-7} + 2.9256 \times 10^{-4} = 2.9277 \times 10^{-4} \]

Calculate \(\Delta c\):

\[ \Delta c = \frac{-4.60 \times 10^{-4} \pm \sqrt{2.9277 \times 10^{-4}}}{2} \]

Calculate the square root:

\[ \sqrt{2.9277 \times 10^{-4}} \approx 0.01711 \]

Substitute back:

\[ \Delta c = \frac{-4.60 \times 10^{-4} \pm 0.01711}{2} \]

Calculate the two possible solutions:

1. \(\Delta c = \frac{-4.60 \times 10^{-4} + 0.01711}{2} \approx 0.008325\)
2. \(\Delta c = \frac{-4.60 \times 10^{-4} - 0.01711}{2}\) (This will be negative and not physically meaningful in this context)

Final Answer