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Which value for the constant c makes z=-(5)/(4) an extraneous solution in the following equation? {:[sqrt(4z+9)=cz+8],[c=]:}

Which value for the constant c c makes z=−54 z=-\frac{5}{4} an extraneous solution in the following equation?\newline4z+9=cz+8c=□ \begin{array}{l} \sqrt{4 z+9}=c z+8 \\ c=\square \end{array}

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Q. Which value for the constant c c makes z=−54 z=-\frac{5}{4} an extraneous solution in the following equation?\newline4z+9=cz+8c=□ \begin{array}{l} \sqrt{4 z+9}=c z+8 \\ c=\square \end{array}
  1. Given equation and extraneous solution: We are given that z=−54z=-\frac{5}{4} is an extraneous solution to the equation 4z+9=cz+8\sqrt{4z+9}=cz+8. To find the value of cc that makes this true, we need to substitute z=−54z=-\frac{5}{4} into the equation and solve for cc.
  2. Substitute and simplify left side: First, substitute z=−54z=-\frac{5}{4} into the left side of the equation 4z+9\sqrt{4z+9}.4(−54)+9=−5+9=4=2\sqrt{4\left(-\frac{5}{4}\right) + 9} = \sqrt{-5 + 9} = \sqrt{4} = 2.
  3. Substitute and simplify right side: Now, substitute z=−54z=-\frac{5}{4} into the right side of the equation cz+8cz+8.c(−54)+8=−5c4+8c\left(-\frac{5}{4}\right) + 8 = -\frac{5c}{4} + 8.
  4. Set expressions equal to find contradiction: Since z=−54z=-\frac{5}{4} is an extraneous solution, the two sides of the equation will not be equal. Therefore, we set the two expressions equal to each other and look for a contradiction.2=−5c4+8.2 = -\frac{5c}{4} + 8.
  5. Isolate term with c: To solve for c, we first isolate the term with c on one side.\newline−5c4=2−8=−6-\frac{5c}{4} = 2 - 8 = -6.
  6. Solve for c: Now, multiply both sides by −45-\frac{4}{5} to solve for c.\newline$c = \left(-\frac{\(4\)}{\(5\)}\right) \times (\(-6\)) = \frac{\(24\)}{\(5\)} = \(4\).\(8\).

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