Math Solutions

Find the remainder when x⁴−3x³+2x−5 is divided by (x−2). The remainder theorem says: R = f(2).

Verify x=3 is a root of x³−6x²+11x−6, then factor completely.

Divide 2x⁴+3x³−x²+5x−4 by x²+2. Show long division steps like arithmetic.

Divide x³−4x²+x+6 by (x−2) using synthetic division. Animate the carry-multiply-add process.

A ball is thrown upward with h(t) = −5t² + 20t + 1. When does it hit the ground? Find max height and total flight time.

Factor x³−27 using the difference of cubes: a³−b³ = (a−b)(a²+ab+b²). Also show a²−b² = (a−b)(a+b).

Factor 2x³+3x²+4x+6 by grouping. Group first two and last two terms, then factor out common binomial.

For ax²+bx+c=0, adjust a, b, c and watch the parabola show 2 roots, 1 root, or 0 real roots based on Δ = b²−4ac.

Solve x²+6x+2=0 by completing the square. Animate building the perfect square geometrically.

Solve 2x²−5x−3=0 using x = (−b ± √(b²−4ac))/(2a). Visualize the parabola and roots.

Factor x² + 7x + 12 = (x + 3)(x + 4). Show the rectangle area model.

Rearrange A = P(1 + rt) to solve for r. Show each algebraic step on a balance scale.

A store sells apples at $2 and oranges at $3. You buy 10 fruits for $24. How many of each?

Solve x+y+z=6, 2x−y+z=3, x+2y−z=2. Show 3D plane intersections converging to a point.

Solve y = −x + 5 and y = 2x − 1 by finding the intersection point on a graph.

Solve 2x + 3y = 12 and 4x − 3y = 6 by elimination. Add the equations to eliminate y.

Solve y = 2x + 1 and 3x + y = 16 using substitution. Show the variable replacement step visually.

Solve (2/3)x − 1/4 = 5/6. Show how to clear fractions by multiplying both sides by the LCD, then solve the resulting integer equation.

Solve 3x + 7 = 22 for x. Visualize each step as maintaining balance on a scale — what you do to one side, you must do to the other.

Maximize P=5x+4y subject to x+y≤10, 2x+y≤14, x≥0, y≥0. Find the optimal corner point.