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Two charges are located on the x axis as shown in the figure. d=10.0 cm, q​1​​ =+6.97 μC, and q​2​​ =−10.4 μC. What charge q​3​​ needs to be placed at position x=3d so that the magnitude of the electric field at the origin is zero?

you want

3
∑qiqj/dij^2 = 0
i,j=1
i≠j
for suitable definitions of dij, which you have not provided

In other words we do not have your figure so we do not really know where the first two charges are located.

To find the charge q3 that needs to be placed at position x=3d such that the magnitude of the electric field at the origin is zero, we can use the principle of superposition.

The electric field at the origin due to multiple charges is the vector sum of the electric fields produced by each individual charge.

Step 1: Calculate the electric field at the origin due to q1.
The electric field E1 due to q1 can be calculated using Coulomb's law,
E1 = k |q1| / r^2,
where k is the electrostatic constant, |q1| is the magnitude of q1, and r is the distance between q1 and the origin (x=0).

Step 2: Calculate the electric field at the origin due to q2.
Similarly, the electric field E2 due to q2 can be calculated,
E2 = k |q2| / r^2,
where |q2| is the magnitude of q2 and r is the distance between q2 and the origin (x=0).

Step 3: Calculate the electric field at the origin due to q3.
The electric field E3 due to q3 can also be calculated using Coulomb's law,
E3 = k |q3| / r^2,
where |q3| is the magnitude of q3 and r is the distance between q3 and the origin (x=0).

Step 4: Set the total electric field at the origin to zero.
To make the magnitude of the electric field at the origin zero, the vector sum of the electric fields due to each charge at the origin must be zero.
E_total = E1 + E2 + E3 = 0

Step 5: Substitute the given values and solve for q3.
Substituting the given values, we have:
E1 = k |q1| / r^2,
E2 = k |q2| / r^2,
E3 = k |q3| / r^2,
E_total = E1 + E2 + E3 = 0.

By substituting the known values of |q1|, |q2|, and the distances r, and solving for |q3|, we can determine the charge q3 that needs to be placed at position x=3d.