To determine the number and type of solutions for the quadratic equation \( 2x^2 - 4x = -2 \), we first need to rearrange it into standard form, which is \( ax^2 + bx + c = 0 \).
Starting with the given equation: \[ 2x^2 - 4x + 2 = 0 \]
Now we identify the coefficients:
- \( a = 2 \)
- \( b = -4 \)
- \( c = 2 \)
Next, we calculate the discriminant \( D \) using the formula: \[ D = b^2 - 4ac \]
Substituting in the values of \( a \), \( b \), and \( c \): \[ D = (-4)^2 - 4(2)(2) \] \[ D = 16 - 16 \] \[ D = 0 \]
The discriminant \( D \) is 0, which indicates that there is exactly one real solution (also called a double root) for the quadratic equation.
In summary:
- The quadratic equation \( 2x^2 - 4x + 2 = 0 \) has one real solution (a double root).
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