Question

what is the discriminant of the quadratic equation 3 - 4x = -6x²?
-68
-56
76
88

Explanation:

Step1: Rewrite in standard form

First, we need to rewrite the quadratic equation \(3 - 4x=-6x^{2}\) in the standard form \(ax^{2}+bx + c = 0\).
By adding \(6x^{2}\) to both sides, we get \(6x^{2}-4x + 3=0\). Here, \(a = 6\), \(b=-4\), and \(c = 3\).

Step2: Recall discriminant formula

The discriminant \(D\) of a quadratic equation \(ax^{2}+bx + c=0\) is given by the formula \(D=b^{2}-4ac\).

Step3: Substitute values into formula

Substitute \(a = 6\), \(b=-4\), and \(c = 3\) into the discriminant formula:
\(D=(-4)^{2}-4\times6\times3\)
First, calculate \((-4)^{2}=16\) and \(4\times6\times3 = 72\).
Then, \(D=16-72=- 56\)? Wait, no, wait, I made a mistake. Wait, the original equation: let's re - check the standard form.
Wait, the original equation is \(3-4x=-6x^{2}\). Let's move all terms to the left side: \(6x^{2}-4x + 3=0\)? Wait, no, if we have \(3-4x=-6x^{2}\), add \(6x^{2}\) to both sides: \(6x^{2}+3 - 4x=0\), which is \(6x^{2}-4x + 3=0\). But wait, let's check the calculation of the discriminant again.
Wait, \(b=-4\), so \(b^{2}=(-4)^{2}=16\), \(4ac = 4\times6\times3=72\), so \(D=16 - 72=-56\)? But wait, maybe I messed up the sign of the equation. Wait, let's start over.
The quadratic equation is \(3-4x=-6x^{2}\). Let's rewrite it as \(6x^{2}-4x + 3 = 0\)? Wait, no, if we add \(6x^{2}\) to both sides, we get \(6x^{2}+3-4x = 0\), or \(6x^{2}-4x + 3=0\). But let's check the discriminant formula again. The discriminant is \(b^{2}-4ac\), where \(a = 6\), \(b=-4\), \(c = 3\). So \(D=(-4)^{2}-4\times6\times3=16 - 72=-56\). Wait, but let's check the equation again. Maybe I made a mistake in the standard form.
Wait, the original equation is \(3-4x=-6x^{2}\). Let's move all terms to the right side: \(0=-6x^{2}+4x - 3\), then multiply both sides by - 1: \(6x^{2}-4x + 3=0\). So \(a = 6\), \(b=-4\), \(c = 3\). Then \(D=b^{2}-4ac=(-4)^{2}-4\times6\times3=16 - 72=-56\).

Answer:

-56 (corresponding to the option with -56)