Algebraic Invariance Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Show that the discriminant b2−4acb^2 - 4ac is invariant under the substitution x=t+kx = t + k in ax2+bx+c=0ax^2 + bx + c = 0.

Solution

  1. 1
    Step 1: Substitute: a(t+k)2+b(t+k)+c=at2+(2ak+b)t+(ak2+bk+c)=0a(t+k)^2 + b(t+k) + c = at^2 + (2ak+b)t + (ak^2+bk+c) = 0.
  2. 2
    Step 2: New coefficients: a′=aa' = a, b′=2ak+bb' = 2ak+b, c′=ak2+bk+cc' = ak^2+bk+c.
  3. 3
    Step 3: New discriminant: (2ak+b)2−4a(ak2+bk+c)=4a2k2+4abk+b2−4a2k2−4abk−4ac=b2−4ac(2ak+b)^2 - 4a(ak^2+bk+c) = 4a^2k^2+4abk+b^2-4a^2k^2-4abk-4ac = b^2-4ac.
  4. 4
    The discriminant is unchanged ✓

Answer

The discriminant b2−4acb^2 - 4ac is preserved.
The discriminant determines the number of real roots, which shouldn't change just because we shift the variable. This invariance confirms that horizontal translation doesn't affect the nature of solutions.

About Algebraic Invariance

Algebraic properties or quantities that remain unchanged when specific algebraic transformations are applied to an expression or system.

Learn more about Algebraic Invariance →

More Algebraic Invariance Examples

Example 1 medium

The polynomial [formula] can be rewritten as [formula]. What is invariant?

Example 3 easy

When you factor [formula], what is invariant?

Example 4 medium

What changes and what stays the same when multiplying both sides of [formula] by 3?

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