Quadratic Formula Derivation using Binomial Square Identity

animations/QuadraticFormulaDerivation.py

@@ -0,0 +1,397 @@
+from manim import *
+from manim.utils.rate_functions import ease_in_cubic, ease_out_sine, ease_in_out_cubic
+from utils.latex_helpers import (derive_equation, DerivationStep, color_tex_by_ranges,
+                                 START_G, END_G)
+
+MOROCCAN_BLUE = ManimColor("#27ADF5")
+
+
+class QuadraticFormulaDerivation(Scene):
+    def construct(self):
+        # --- Title
+
+        self.wait(1)
+
+        self.play(
+            Write(Text(
+                "Derivation of the Quadratic Formula",
+                font_size=36
+            ).to_edge(UP)),
+            run_time=1.25
+        )
+
+        self.wait(1)
+
+        # --- Phase 1: Present the quadratic equation, then convert the left hand to monic form
+
+        equation = MathTex("a", "x^2", "+", "b", "x", "+", "c", "=", "0")
+
+        self.play(Write(equation, run_time=1.5))
+        equation = derive_equation(self, equation, [
+            # We need to keep TeX groups separate so TransformMatchingTex animation doesn't turn into a fade.
+            # Hence, we can't use \frac for fractions since they require grouped {numerator}{denominator} in a single TeX group.
+            # We also can't use \over, since the = sign and the right side of the equation would fall into the denominator.
+            # A feasible option is delimiting numerator and denominator with \left. and \right.
+
+            # Alternate between a and {a} when it's not supposed to move between terms.
+
+            DerivationStep(
+                START_G, "a", "x^2", "+", "b", "x", "+", "c", r"\over a", END_G,
+                START_G, "=", END_G,
+                START_G, "0", r"\over a", END_G
+            ),
+            DerivationStep(
+                "x^2", "+", START_G, "b", r"\over {a}", END_G, "x", "+", START_G, "c", r"\over {a}", END_G,
+                START_G, "=", END_G,
+                "0",
+                delay=.75, transition_duration=.75
+            )
+        ])
+
+        self.wait(1)
+
+        # --- Phase 2: Analytically prove square completion using the binomial square identity
+
+        # -- Present the binomial square identity in the form (x+p)² = x² + 2px + p²
+
+        self.play(
+            equation.animate.shift(2 * UP)
+        )
+
+        self.wait(.5)
+
+        bsi_title = Text("Binomial Square Identity", font_size=36)
+        bsi_title.move_to(5 * DOWN)
+
+        bsi_formula = MathTex("(x +", "p", ")^2 =", "x^2", "+", "2", "p", "x", "+", "p", "^2")
+        bsi_formula.add_updater(lambda m: m.next_to(bsi_title, DOWN, buff=0.5))
+
+        self.add(bsi_title, bsi_formula)
+        self.play(
+            bsi_title.animate.move_to(.5 * UP),
+            run_time=1.5
+        )
+
+        self.wait(.75)
+        self.play(FadeOut(bsi_title))
+        bsi_formula.clear_updaters()
+
+        self.wait(.75)
+
+        # -- Replace p with b / a in binomial square identity
+
+        # Parentheses from the original equation don't change size at first, so the animation looks cleaner
+        bsi_formula_replaced = MathTex(
+            "(x +", START_G, "b", r"\over {a}", END_G, ")^2 =", "x^2", "+", "2", START_G, "b", r"\over {a}",
+            END_G, "x", "+", r"\bigl(", START_G, "b", r"\over {a}", END_G, r"\bigr)", "^2"
+        ).move_to(bsi_formula)
+
+        p_indices = [(1, 5), (10, 13), (17, 20)]
+
+        equation[2:6].set_color(ORANGE) # b / a
+        b_over_a_copies = {
+            (start, end): equation[2:6].copy()
+            for (start, end) in p_indices
+        }
+
+        self.play(
+            AnimationGroup(
+                Indicate(equation[2:6], color=ORANGE, run_time=.5),
+                AnimationGroup(
+                    AnimationGroup(
+                        *[
+                            Transform(
+                                b_over_a_copies[start, end],
+                                bsi_formula_replaced[start:end].set_color(ORANGE)
+                            )
+                            for start, end in p_indices
+                        ]
+                    ),
+                    TransformMatchingShapes(bsi_formula, bsi_formula_replaced, run_time=.65),
+                    lag_ratio=.45
+                ),
+                lag_ratio=.35
+            )
+        )
+
+        self.remove(bsi_formula, *b_over_a_copies.values())
+        bsi_formula = bsi_formula_replaced
+        self.add(bsi_formula)
+
+        # Make initial parentheses bigger since they're now wrapping a fraction
+        bsi_formula_replaced = MathTex(
+            r"\bigl(x +", START_G, "b", r"\over {a}", END_G, r"\bigr)^2 =", "x^2", "+", "2", START_G, "{b}",
+            r"\over {a}", END_G, "x", "+", r"\bigl(", START_G, "{{b}}", r"\over {a}", END_G, r"\bigr)", "^2"
+        ).move_to(bsi_formula)
+        for b_over_a_index in (replaced_b_over_a_indices := [2, 3, 10, 11, 17, 18]):
+            bsi_formula_replaced[b_over_a_index].set_color(ORANGE)
+
+        self.play(bsi_formula.animate.become(bsi_formula_replaced))
+
+        self.wait(.5)
+
+        # -- Replace b / a with b / 2a in binomial square identity
+
+        # Separate denominators from the fractions
+        # Make multiplying 2 phantom so TransformMatchingShapes doesn't copy it to the new denominators
+        two_copy = bsi_formula[8].copy()
+        bsi_formula.become(
+            MathTex(
+                r"\bigl(x +", START_G, "b", r"\over", "{a}", END_G, r"\bigr)^2 =", "x^2", "+", r"\phantom{2}", START_G, "{b}",
+                r"\over", "{a}", END_G, "x", "+", r"\bigl(", START_G, "{{b}}", r"\over", "{a}", END_G, r"\bigr)", "^2"
+            ).move_to(bsi_formula)
+        )
+        for start, end in [(1, 5), (10, 14), (18, 22)]:
+            bsi_formula[start:end].set_color(ORANGE)
+
+        bsi_formula_replaced = MathTex(
+            r"\bigl(x +", START_G, "b", r"\over {2a}", END_G, r"\bigr)^2 =", "x^2", "+", "2", START_G, "b", r"\over {2a}",
+            END_G, "x", "+", r"\bigl(", START_G, "b", r"\over {2a}", END_G, r"\bigr)", "^2"
+        ).move_to(bsi_formula)
+        for b_over_a_index in replaced_b_over_a_indices:
+            bsi_formula_replaced[b_over_a_index].set_color(YELLOW_E)
+
+        two_final = bsi_formula_replaced[8]
+
+        self.play(
+            # Separate it in two transformations so initial parentheses aren't copied to the final ones
+            TransformMatchingShapes(bsi_formula[0], replaced := bsi_formula_replaced[0]),
+            TransformMatchingShapes(bsi_formula[1:], replaced1 := bsi_formula_replaced[1:]),
+
+            Transform(two_copy, two_final)
+        )
+
+        bsi_formula_replaced = MathTex(
+            r"\bigl(x +", START_G, "b", r"\over {2a}", END_G, r"\bigr)^2 =", "x^2", "+", r"\phantom{2}", START_G,
+            "b", r"\over {2a}", END_G, "x", "+", r"\bigl(", START_G, "b", r"\over {2a}", END_G, r"\bigr)", "^2"
+        ).move_to(bsi_formula)
+        for start, end in [(1, 5), (9, 13), (16, 20)]:
+            bsi_formula_replaced[start:end].set_color(YELLOW_E)
+
+        self.remove(replaced, replaced1)
+        self.add(bsi_formula.become(bsi_formula_replaced))
+
+        self.wait(1)
+
+        # -- Cancel twos in 2(b / 2a), in binomial square identity
+
+        bsi_formula_replaced = MathTex(
+            r"\bigl(x +", START_G, "b", r"\over", "{2a}", END_G, r"\bigr)^2 =", "x^2", "+", START_G, "b",
+            r"\over", "{a}", END_G, "x", "+", r"\bigl(", START_G, "b", r"\over", "{2a}", END_G, r"\bigr)",
+            "^2"
+        ).move_to(bsi_formula)
+
+        bsi_formula_replaced[1:6].set_color(YELLOW_E)
+        bsi_formula_replaced[9:14].set_color(ORANGE)
+        bsi_formula_replaced[17:22].set_color(YELLOW_E)
+
+        denominator_copies = [bsi_formula_replaced[index].copy() for index in [4, 12, 20]]
+        for denominator in denominator_copies:
+            denominator.shift(.1 * DOWN)
+
+        bsi_formula_replaced1 = MathTex(
+            r"\bigl(x +", START_G, "b", r"\over \phantom{{2a}}", END_G, r"\bigr)^2 =", "x^2", "+", START_G, "b",
+            r"\over \phantom{{a}}", END_G, "x", "+", r"\bigl(", START_G, "b", r"\over \phantom{{2a}}", END_G, r"\bigr)", "^2"
+        ).move_to(bsi_formula)
+
+        bsi_formula_replaced1[1:5].set_color(YELLOW_E)
+        bsi_formula_replaced1[9:11].set_color(ORANGE)
+        bsi_formula_replaced1[16:18].set_color(YELLOW_E)
+
+        self.play(
+            FadeIn(*denominator_copies, run_time=.75),
+            FadeOut(two_copy, run_time=.75),
+            TransformMatchingShapes(bsi_formula, bsi_formula_replaced1)
+        )
+
+        self.remove(*denominator_copies, bsi_formula_replaced1)
+        self.add(bsi_formula.become(bsi_formula_replaced).shift(.1 * DOWN))
+
+        # -- (b / 2a)² to left hand of binomial square identity
+
+        bsi_formula = derive_equation(self, bsi_formula, [
+            DerivationStep( # Reformat TeX
+                r"\bigl(x +", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2", "=",
+                "x^2 +", (rf"{START_G} b \over a {END_G}", ORANGE), "x",
+                "+", r"{\bigl(}", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                delay=0, transition_duration=.25, transform_type=FadeTransform
+            ),
+            DerivationStep( # Add -(b / a)² to both hands
+                r"\bigl(x +", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                "-", r"\bigl(", (rf"{{{START_G} b \over 2a {END_G}}}", YELLOW_E), r"{\bigr)^2}", "=",
+                "x^2 +", (rf"{START_G} b \over a {END_G}", ORANGE), "x"
+                "+", r"{\bigl(}", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                "-", r"{\bigl(}", (rf"{{{{{START_G} b \over 2a {END_G}}}}}", YELLOW_E), r"{{\bigr)^2}}",
+                delay=.75, transition_duration=1.25
+            ),
+            DerivationStep( # Color cancelling terms in red
+                r"\bigl(x +", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                "-", r"\bigl(", (rf"{{{START_G} b \over 2a {END_G}}}", YELLOW_E), r"{\bigr)^2}", "=",
+                "x^2 +", (rf"{START_G} b \over a {END_G}", ORANGE), "x"
+                "+", r"{\bigl(}", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                "-", r"{\bigl(}", rf"{{{{{START_G} b \over 2a {END_G}}}}}", r"{{\bigr)^2}}",
+                on_build=lambda tex: tex[11:].set_color(RED_C),
+                rate_func=ease_out_sine, transition_duration=.5
+            ),
+            DerivationStep( # Remove cancelling terms
+                r"\bigl(x +", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                "-", r"\bigl(", (rf"{{{START_G} b \over 2a {END_G}}}", YELLOW_E), r"\bigr)^2", "=",
+                "x^2 +", (rf"{START_G} b \over a {END_G}", ORANGE), "x",
+                delay=.5
+            )
+        ])
+
+        # -- Left hand of binomial square identity to quadratic equation
+
+        self.wait(.5)
+        self.play(
+            bsi_formula[:7].animate.set_color(GREEN),
+            bsi_formula[8:].animate.set_color(MOROCCAN_BLUE),
+            equation[:7].animate.set_color(MOROCCAN_BLUE),
+            run_time=1.5, rate_func=ease_in_cubic
+        )
+
+        equation = derive_equation(self, equation, [
+            DerivationStep(
+                r"\bigl(x +", (rf"{START_G} b \over 2a {END_G}", YELLOW_E), r"\bigr)^2",
+                "-", r"\bigl(", (rf"{{{START_G} b \over 2a {END_G}}}", YELLOW_E), r"\bigr)^2",
+                "+", START_G, "c", r"\over {a}", END_G, START_G, "=", END_G, "0",
+                on_build=lambda tex: tex[:7].set_color(GREEN),
+                delay = 1.5, transform_type=TransformMatchingShapes
+            )
+        ])
+
+        # -- Fade out binomial square identity, equation back to focus
+
+        self.wait(1.5)
+        self.play(
+            LaggedStart(
+                FadeOut(bsi_formula),
+                equation.animate.shift(2 * DOWN) \
+                    .set_color(WHITE),
+                lag_ratio=.5
+            )
+        )
+
+        # --- Phase 3: Isolate x and simplify expression
+
+        equation = derive_equation(self, equation, [
+            DerivationStep( # Add (b / 2a)² to both hands
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                "-", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                "+", START_G, "c", r"\over {a}", END_G, START_G,
+                "+", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                "=", END_G, "0",
+                "+", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                delay=1.5
+            ),
+            DerivationStep( # Color cancelling terms in red
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                "-", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                "+", START_G, "c", r"\over {a}", END_G, START_G,
+                "+", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                "=", END_G, "0",
+                "+", r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                on_build=lambda tex: color_tex_by_ranges(tex, RED_C, (3,7), (13,17)),
+                rate_func=ease_out_sine, transition_duration=.5, delay=.5
+            ),
+            DerivationStep( # Remove cancelling terms
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                "+", START_G, "c", r"\over {a}", END_G,
+                START_G, "=", END_G,
+                r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                delay=.5
+            ),
+            DerivationStep( # Subtract c / a to both hands
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                "+", START_G, "c", r"\over {a}", END_G,
+                rf"- {START_G} c \over {{a}} {END_G}",
+                START_G, "=", END_G,
+                r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                rf"{{- {START_G} c \over {{a}} {END_G}}}"
+            ),
+            DerivationStep( # Color cancelling terms in red
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                ("+", RED_C), START_G, ("c", RED_C), (r"\over {a}", RED_C), END_G,
+                (rf"- {START_G} c \over {{a}} {END_G}", RED_C),
+                START_G, "=", END_G,
+                r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                rf"{{- {START_G} c \over {{a}} {END_G}}}",
+                rate_func=ease_out_sine, transition_duration=.5, delay=.5
+            ),
+            DerivationStep( # Remove cancelling terms
+                r"\bigl(x +", rf"{START_G} b \over 2a {END_G}", r"\bigr)^2",
+                START_G, "=", END_G,
+                r"\bigl(", rf"{{{START_G} b \over 2a {END_G}}}", r"\bigr)^2",
+                rf"{{- {START_G} c \over {{a}} {END_G}}}",
+                delay=.5
+            ),
+            DerivationStep( # (b / 2a)² ⟶ b² / 4a²
+                r"{\bigl(}x +", rf"{START_G} b \over 2a {END_G}", r"{\bigr)}^2",
+                START_G, "=", END_G,
+                rf"{{{START_G} b^2 \over 4a^2 {END_G}}}",
+                rf"{{- {START_G} c \over {{a}} {END_G}}}"
+            ),
+            DerivationStep( # Combine the two fractions in right hand
+                r"{\bigl(}x +", rf"{START_G} b \over 2a {END_G}", r"{\bigr)}^2",
+                START_G, "=", END_G,
+                r"\frac{b^2 - 4ac}{4a^2}"
+            ),
+            DerivationStep( # Take square out of left hand
+                "x", "+", START_G, "b" r"\over", "2a", END_G,
+                START_G, "=", END_G,
+                r"\pm \sqrt \frac{b^2 - 4ac}{4a^2}",
+                transform_type=FadeTransform
+            ),
+            DerivationStep( # Simplify square root from right hand
+                "x", "+", START_G, "b" r"\over", "2a", END_G,
+                START_G, "=", END_G,
+                r"\pm", START_G, r"\sqrt{b^2 - 4ac}", r"\over 2a", END_G
+            ),
+            DerivationStep( # ± to right hand numerator
+                "x", "+", START_G, "b" r"\over", "2a", END_G,
+                START_G, "=", END_G,
+                START_G, r"\pm", r"\sqrt{b^2 - 4ac}", r"\over 2a", END_G
+            ),
+            DerivationStep( # Subtract b / 2a from both hands
+                "x", "+", START_G, "b" r"\over", "2a", END_G,
+                "{-}", START_G, "{b}" r"\over", "{2a}", END_G,
+                START_G, "=", END_G,
+                START_G, r"\pm", r"\sqrt{b^2 - 4ac}", r"\over 2a", END_G,
+                "-", START_G, "b" r"\over", "2a", END_G,
+            ),
+            DerivationStep( # Color cancelling terms in red
+                "x", "+", START_G, "b" r"\over", "2a", END_G,
+                "{-}", START_G, "{b}" r"\over", "{2a}", END_G,
+                START_G, "=", END_G,
+                START_G, r"\pm", r"\sqrt{b^2 - 4ac}", r"\over 2a", END_G,
+                "-", START_G, "b" r"\over", "2a", END_G,
+                on_build=lambda tex: tex[1:12].set_color(RED_C),
+                rate_func=ease_out_sine, transition_duration=.5, delay=.5
+            ),
+            DerivationStep( # Remove cancelling terms
+                "x", START_G, "=", END_G,
+                START_G, r"\pm", r"\sqrt{b^2 - 4ac}", r"\over 2a", END_G,
+                "-", START_G, "b" r"\over", "2a", END_G,
+                delay=.5
+            ),
+            DerivationStep( # Join remaining fractions in right hand
+                "x", START_G, "=", END_G,
+                START_G, r"\pm", r"\sqrt{b^2 - 4ac}", "-", "b", r"\over", "2a", END_G
+            ),
+            DerivationStep( # Reorder right hand numerator terms
+                "x", START_G, "=", END_G,
+                START_G, "-", "b", r"\pm", r"\sqrt{b^2 - 4ac}", r"\over", "2a", END_G
+            )
+        ])
+
+        self.wait(1)
+
+        # --- Show off result
+
+        self.play(Create(
+            SurroundingRectangle(equation, color=YELLOW_E, buff=.25),
+            rate_func=ease_in_out_cubic
+        ))
+
+        self.wait(1.5)