For 4x squared plus 6xy plus 2y squared minus 7 equals 0, what does the discriminant show?

1. Read the coefficients from the general form: A is 4, B is 6 and C is 2. 2. The discriminant is B squared minus 4AC. 3. B squared is 36. 4. The product 4AC is 4 times 4 times 2, which is 32. 5. Subtracting gives 36 minus 32, so the discriminant is 4. 6. A positive discriminant marks a hyperbola, a zero one a parabola and a negative one an ellipse. 7. So the curve is a hyperbola. 8. Reading it as an ellipse ignores that the value came out positive rather than negative. 9. The value 68 comes from adding 4AC instead of subtracting it. 10. The value 0 would need B squared to match 4AC exactly, which it does not. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.4 Rotation of Axes_