To determine the values of m for which the equation mx^2 - 2mx + 3 = 0 has two solutions for x, we can use the discriminant. The discriminant is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

In this case, a = m, b = -2m, and c = 3. Substituting these values into the discriminant formula, we have (-2m)^2 - 4(m)(3).

Simplifying further, we get 4m^2 - 12m.

For the equation to have two solutions, the discriminant must be greater than zero. So we set the discriminant greater than zero and solve the inequality:

4m^2 - 12m > 0

Factoring out 4m, we have 4m(m - 3) > 0.

Now we have two factors: 4m and (m - 3). To determine the values of m that satisfy the inequality, we can use the sign chart method.

We set each factor equal to zero and solve for m:

4m = 0 --> m = 0 m - 3 = 0 --> m = 3

Now we have two critical points: m = 0 and m = 3. We can use these points to create a sign chart:

-∞ | 0 | 3 | +∞

(+) | (-) | (+) | (+)

From the sign chart, we can see that the inequality is satisfied when m is less than 0 or when m is greater than 3. Therefore, the values of m for which the equation has two solutions for x are m < 0 and m > 3.

This problem has been solved

Question

To determine the values of m for which the equation mx^2 - 2mx + 3 = 0 has two solutions for x, we can use the discriminant. The discriminant is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

In this case, a = m, b = -2m, and c = 3. Substituting these values into the discriminant formula, we have (-2m)^2 - 4(m)(3).

Simplifying further, we get 4m^2 - 12m.

For the equation to have two solutions, the discriminant must be greater than zero. So we set the discriminant greater than zero and solve the inequality:

4m^2 - 12m > 0

Factoring out 4m, we have 4m(m - 3) > 0.

Now we have two factors: 4m and (m - 3). To determine the values of m that satisfy the inequality, we can use the sign chart method.

We set each factor equal to zero and solve for m:

4m = 0   -->   m = 0
m - 3 = 0   -->   m = 3

Now we have two critical points: m = 0 and m = 3. We can use these points to create a sign chart:

-∞    |    0    |    3    |    +∞
_____________________________________
   (+)   |   (-)   |   (+)   |   (+)

From the sign chart, we can see that the inequality is satisfied when m is less than 0 or when m is greater than 3. Therefore, the values of m for which the equation has two solutions for x are m < 0 and m > 3.

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