Question

homework 9: section 2.5
score: 235/280 answered: 24/28
question 26
solve equation by completing the square. list the solutions, separated by commas.
$4z^2 + 3z - 5 = 0$
$z = $
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Explanation:

Step1: Divide by coefficient of \(z^2\)

Divide the entire equation \(4z^{2}+3z - 5=0\) by \(4\) to make the coefficient of \(z^{2}\) equal to \(1\).
We get \(z^{2}+\frac{3}{4}z-\frac{5}{4}=0\).

Step2: Move constant term to right

Rearrange the equation to get \(z^{2}+\frac{3}{4}z=\frac{5}{4}\).

Step3: Complete the square

Take half of the coefficient of \(z\), which is \(\frac{3}{8}\), square it to get \((\frac{3}{8})^{2}=\frac{9}{64}\). Add this to both sides of the equation:
\(z^{2}+\frac{3}{4}z+\frac{9}{64}=\frac{5}{4}+\frac{9}{64}\).
The left side is now a perfect square: \((z + \frac{3}{8})^{2}=\frac{5\times16}{4\times16}+\frac{9}{64}=\frac{80}{64}+\frac{9}{64}=\frac{89}{64}\).

Step4: Solve for \(z\)

Take the square root of both sides: \(z+\frac{3}{8}=\pm\sqrt{\frac{89}{64}}=\pm\frac{\sqrt{89}}{8}\).
Then solve for \(z\): \(z=-\frac{3}{8}\pm\frac{\sqrt{89}}{8}=\frac{- 3\pm\sqrt{89}}{8}\).

Answer:

\(\frac{-3 + \sqrt{89}}{8},\frac{-3 - \sqrt{89}}{8}\)