Question

Which value for the constant c makes z =-5/4 an extraneous solution in the following equation?

√(4z + 9) = cz + 8 ?
a) c = -2
b) c= -1
c) c= 0
d) c= 1

Answer 1

None of the values provided for constant c (-2, -1, 0, 1) make z = -5/4 an extraneous solution to the equation. Substituting z into the equation and solving for c results in a value not present in the options, which is c = 24/5.

Step-by-step explanation:

The student's question is asking which value of the constant c makes z = -5/4 an extraneous solution for the equation √(4z + 9) = cz + 8. To find this, we substitute z = -5/4 into the equation to see which value of c will result in an equation that cannot be true.

Substituting z gives us:

• √(4(-5/4) + 9) = c(-5/4) + 8
• √(-5 + 9) = -5c/4 + 8
• √(4) = -5c/4 + 8
• 2 = -5c/4 + 8

Now we solve for c:

1. Multiply both sides by 4:
2. 8 = -5c + 32
3. Subtract 32 from both sides:
4. -24 = -5c
5. Divide by -5:
6. c = 24/5, which is not one of the provided options, meaning none of the given values for c make the equation untrue when z = -5/4.

Therefore, there is no valid answer from the provided options (a) c = -2, (b) c = -1, (c) c = 0, or (d) c = 1 to make z = -5/4 an extraneous solution for the given equation.