To find the discriminant of the quadratic equation, we first need to get the equation into standard form, which is [tex]\( ax^2 + bx + c = 0 \)[/tex].
Given the equation:
[tex]\[ 3 - 4x = -6x^2 \][/tex]
We need to rearrange this equation to standard form:
[tex]\[ -6x^2 - 4x + 3 = 0 \][/tex]
Here, the coefficients are:
- [tex]\( a = -6 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the coefficients:
[tex]\[ \Delta = (-4)^2 - 4 \cdot (-6) \cdot 3 \][/tex]
Calculating step by step:
1. Calculate [tex]\( (-4)^2 \)[/tex]:
[tex]\[ (-4)^2 = 16 \][/tex]
2. Calculate [tex]\( 4 \cdot (-6) \cdot 3 \)[/tex]:
[tex]\[ 4 \cdot (-6) = -24 \][/tex]
[tex]\[ -24 \cdot 3 = -72 \][/tex]
3. Subtract the second result from the first:
[tex]\[ \Delta = 16 - (-72) \][/tex]
Since subtracting a negative is the same as adding:
[tex]\[ \Delta = 16 + 72 \][/tex]
[tex]\[ \Delta = 88 \][/tex]
Thus, the discriminant of the quadratic equation [tex]\( 3 - 4x = -6x^2 \)[/tex] is [tex]\( 88 \)[/tex]. The correct answer is:
[tex]\[ 88 \][/tex]
