Question

what is the value of the discriminant for the quadratic equation $-3 = -x^2 + 2x$?
discriminant $= b^2 - 4ac$
\bigcirc $-8$
\bigcirc $4$
\bigcirc $8$
\bigcirc $16

Explanation:

Step1: Rewrite the equation in standard form

First, we need to rewrite the quadratic equation \(-3 = -x^2 + 2x\) in the standard form \(ax^2 + bx + c = 0\).

Add \(x^2\) to both sides and subtract \(2x\) from both sides, then add \(3\) to both sides:

\(x^2 - 2x + 3 = 0\)

Here, \(a = 1\), \(b = -2\), and \(c = 3\).

Step2: Calculate the discriminant

The formula for the discriminant is \(D = b^2 - 4ac\).

Substitute \(a = 1\), \(b = -2\), and \(c = 3\) into the formula:

\(D = (-2)^2 - 4(1)(3)\)

First, calculate \((-2)^2 = 4\).

Then, calculate \(4(1)(3) = 12\).

Now, subtract: \(D = 4 - 12 = -8\)? Wait, that can't be right. Wait, maybe I made a mistake in rewriting the equation.

Wait, let's check again. The original equation is \(-3 = -x^2 + 2x\). Let's move all terms to the left side:

\(x^2 - 2x + 3 = 0\)? Wait, no. Wait, if we add \(x^2\) to both sides: \(x^2 - 3 = 2x\), then subtract \(2x\) from both sides: \(x^2 - 2x - 3 = 0\)? Wait, no, original equation: \(-3 = -x^2 + 2x\). Let's add \(x^2\) to both sides: \(x^2 - 3 = 2x\), then subtract \(2x\) and add \(3\): \(x^2 - 2x + 3 = 0\)? Wait, no, that's incorrect. Wait, let's do it step by step.

Original equation: \(-3 = -x^2 + 2x\)

Add \(x^2\) to both sides: \(x^2 - 3 = 2x\)

Subtract \(2x\) from both sides: \(x^2 - 2x - 3 = 0\)? Wait, no, if we have \(x^2 - 3 = 2x\), then subtract \(2x\) and add \(3\): \(x^2 - 2x + 3 = 0\)? Wait, no, that's not right. Wait, let's move all terms to the right side:

\(0 = -x^2 + 2x + 3\), then multiply both sides by -1: \(x^2 - 2x - 3 = 0\). Ah! There we go. I made a mistake earlier. So the correct standard form is \(x^2 - 2x - 3 = 0\). So \(a = 1\), \(b = -2\), \(c = -3\).

Now, calculate the discriminant: \(D = b^2 - 4ac = (-2)^2 - 4(1)(-3)\)

\((-2)^2 = 4\)

\(4(1)(-3) = -12\)

So \(D = 4 - (-12) = 4 + 12 = 16\). Ah, that's one of the options. So where did I go wrong earlier? I incorrectly moved the terms. Let's redo the equation:

Original equation: \(-3 = -x^2 + 2x\)

Let's rearrange to standard form \(ax^2 + bx + c = 0\).

Add \(x^2\) to both sides: \(x^2 - 3 = 2x\)

Subtract \(2x\) from both sides: \(x^2 - 2x - 3 = 0\)

Yes, that's correct. So \(a = 1\), \(b = -2\), \(c = -3\).

Now, discriminant \(D = b^2 - 4ac = (-2)^2 - 4(1)(-3) = 4 + 12 = 16\).

Ah, that's the correct calculation. So my initial mistake was in the sign of \(c\). So the correct discriminant is 16.

Answer:

16