Math Solutions

Expand (x+y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴. Show coefficients from Pascal's triangle row 4.

Test f(x)=x³ (odd, origin symmetry) and g(x)=x² (even, y-axis symmetry).

Solve x²−4x−5 > 0. Show the parabola with positive regions highlighted and number line below.

Compare y=3x (direct: doubling x doubles y) vs y=12/x (inverse: doubling x halves y).

Compute Σ(k=1 to n) k = n(n+1)/2. Show the triangular number pattern visually.

Visualize 1 + 1/2 + 1/4 + 1/8 + ... converging to 2. Show shrinking rectangles filling a fixed area.

Visualize the sum 3+7+11+15+...+39 as stacking rectangles. Show Gauss's pairing trick.

Graph y≤x+4, y≥−x+2, x≥0. Find the feasible region where all constraints overlap.

Graph y > 2x − 3 and shade the solution region above the line.

Find and graph the inverse of f(x)=3x−2. Show f and f⁻¹ as reflections across y=x.

If f(x)=x² and g(x)=x+3, visualize f(g(x))=(x+3)² vs g(f(x))=x²+3. Show they are different!

Graph f(x) = {x² if x<0, 2x+1 if x≥0} and check continuity at the boundary.

Graph y=(x+1)/(x−2) showing vertical asymptote at x=2 and horizontal asymptote at y=1.

Factor x³−6x²+11x−6 = (x−1)(x−2)(x−3). Show the three roots on the graph.

Show how the leading coefficient and degree determine end behavior. Toggle even/odd degree and positive/negative leading coefficient to see all 4 cases.

Visualize log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(a^n) = n·log(a).

Show y=2^x and y=log₂(x) as reflections across y=x. The log "undoes" the exponential.

Compare y=2^x (growth) vs y=(1/2)^x (decay). Show population doubling and radioactive half-life on the same axes.

Graph y=|x|, y=|x−3|+2, y=−2|x+1|. Show how h, k, a shift and stretch the V-shape.

Show parallel lines (no solution) vs overlapping lines (infinite solutions). Adjust slopes to see when lines become parallel or identical.