Solved math problems

Use two simple equations to find the silver-coin loss after 50 copper coins are gained through exchanges.

Use x²-x=1 to evaluate x⁴-2x³+x² directly, without calculating either root of the quadratic.

Factor (x+1)³-3(x+1)+2 by replacing the repeated expression with a temporary variable.

Group six terms to expose a common quadratic, then factor the remaining expression in x and y.

Use a difference of squares and coefficient comparison to factor x⁴-3x²+1 and x⁴+x²+25.

Factor a square-root equation, keep its domain, and count distinct intersections with a parameterized line.

Compare adjacent terms in two binomial expansions to locate the largest coefficient without expanding every power.

Use P(1) and P(-1) to find the sum of every second coefficient without expanding a sixteenth-degree polynomial.

Identify the hidden products in a long expression and use the square-of-a-sum identity to factor it.

Set x = 4 in f(x,y) = 3x² - 4xy + 5y², then solve the resulting quadratic equation for both possible y-values.

Handle the exceptional parameter values before dividing by a coefficient that may be zero.

Rearrange each equation to zero, factor it, and use the zero-product property to find every root.

Three worked examples show how to factor a quadratic inequality, select intervals, and recognize when there are no real solutions.

Isolate the square, take both signs of the root, and use the imaginary unit when the right side is negative.

Turn two purchase conditions into linear equations, solve for each unit price, then check both original statements.

Two worked limits show how to remove a 0/0 form using polynomial factors or a conjugate expression.

Differentiate a cubic, solve for its critical points, classify the turning points and write the increasing and decreasing intervals.

An expression P dx + Q dy is an exact differential when ∂P/∂y = ∂Q/∂x. Worked example with P = 2x(1 − e^y)/(1 + x²)² and Q = e^y/(1 + x²) + 1.

Parametrise the segment, substitute x, y, dx, dy and integrate over the parameter. Worked example: ∫ 2(x² + y²)dx + (x + y)²dy from B(2; 2) to A(1; 1) equals −56/3.

The gradient is the vector of partial derivatives. Worked example: u = z²/(xy²) at M(1/3, 2, √(2/3)) gives grad u = (−1.5, −0.5, √6/2).

For y = x² - 9, express the distance to (0, 0) as a function of x and find where that distance is smallest.

Differentiate a cubic curve, set its derivative equal to -4, solve for x, then return to the original equation to find both points.

Cut equal squares from the corners of a 15-inch cardboard square. Build V(x) = x(15 - 2x)^2 and find the maximum volume.

For a circle of radius 6, use x² + y² = 36 to write the rectangle area as 4x√(36 - x²) and find its maximum.

Compare corresponding vertices to recognize a 90-degree counterclockwise rotation followed by a two-unit translation.

Combine two conditional statements about congruence and perimeter into one valid geometric conclusion.

Multiply nonzero residues by a number coprime to a prime, then cancel their product modulo that prime.

Distinct powers of two set distinct binary digits. Use exponent positions to count the ones without writing a huge number.

A multiple of five differs from its four-digit reverse by 3627. Use place value and divisibility to find one example.

Use parity and residues modulo four to prove that a Diophantine equation has just one prime-number solution.

Count nonnegative solutions of x + 2y + 5z = 200000 by fixing the number of 5-unit coins, then summing the possible 2-unit coins.

Count two types of handshakes and show why the number of people present must be a perfect square.

Prove a divisibility property by splitting p squared minus one and checking factors of 3 and 8.

Multiply the wall area by the number of coats, divide by coverage, round up to whole litres, then calculate the cost.

If 23.8% of a group is 49,200, divide by 0.238 to recover the total and round only at the end.

Evaluate the function at both endpoints, find the average rate of change, and use it as the slope of the secant line. Worked quadratic example.

Use vertex form and one additional point to reconstruct a parabola's equation.

Complete the square in the exponent of 4^(x²-6x+12) to find its minimum value.

Find every breakpoint in f and g, split the number line into intervals, then add the correct formulas on each interval.

With f(x) = √(3x) and g(x) = 5x − 3: the four combinations and, more importantly, their domains - the root forces x ≥ 0 and the quotient also drops x = 3/5.

Use only the leading term to decide whether a graph rises or falls on its left and right ends.

Combine the formulas, then intersect the domains and throw out the zeros of the denominator. Worked example with f(x) = x² − 16 and g(x) = x² + 6x + 8, including the cancelled-factor trap.

Read peaks, valleys and endpoint values correctly. A worked graph distinguishes local extrema from the absolute highest and lowest values.

Find a function's domain by excluding zero denominators and negative even-root arguments. Four worked examples include a quadratic under a square root.

Find the slope from two points, substitute one point to get the y-intercept, and write the line in slope-intercept form. Includes integer and fractional slopes.

A phone plan graph: flat $40 up to 5 GB, then a straight climb to $290 at 30 GB. Reading values, interpreting them in words, finding the rate of $10 per gigabyte and writing the piecewise formula.

Everything a graph question asks, on one picture: f(−14) and f(−6), the domain and range, the intercepts, where f(x) > 0, and how many times the lines y = 1 and x = 2 cross the graph.

Graph each piece only on its own interval and mark the boundary point: filled dot where the value is included, open dot where it is not. Worked example with g(x) = x + 2 for x ≤ 0 and −x − 1 for x > 0.

Check exponents to distinguish polynomial functions from reciprocal and square-root expressions.

Replace x by -x and compare: f(-x) = f(x) means even, f(-x) = -f(x) means odd. Two worked examples show the common sign traps.

Find every zero of a factored polynomial and classify it by the exponent of its factor.

Start with y = √x, reflect across the x-axis, shift up 8 and right 9 to obtain the final formula and domain.

Two examples: a right triangle whose vertex lies on y = x⁸, and a circle made from a wire of length 5x.

Double the angle made by a bisector, then subtract from 180 degrees to find the adjacent angle.

The minute hand moves 12 times as fast as the hour hand. Convert the hour hand's minute-mark position by multiplying by 12 and reduce modulo 60.

Explain why adjacent 20-degree and 15-degree angles form a 35-degree angle and identify the proof's reason.

Two semicircular cutouts of diameter 16 cm form one circle. Subtract its area from a 31 cm by 16 cm rectangle.

A plane through a top edge and the midpoint of a vertical edge cuts a rectangular prism in a rectangle. Find its area using the Pythagorean theorem.

A drawing uses 1 cm to represent 5 m. Convert both room dimensions before multiplying to find the real area.

Circumference C = 2πr, area A = πr². Two worked examples and a derivation of the area formula by rearranging circular sectors.

Split a composite figure into a rectangle and two triangles, find each area, and add them. A 3 by 6 rectangle with two triangles of base 3 and height 2 has area 24 square units.

A rectangle with a semicircle on each short end has area LW + π(W/2)². Worked example: an 85 m by 52 m rectangle.

Add the two parallel bases, halve the sum, multiply by the height: A = (b₁ + b₂)/2 × h. Two worked examples from real worksheets, including the slanted side that is there to distract you.

Subtract the inner circle from the outer circle: A = π(R² − r²). Worked example with outer radius 9 cm and inner radius 6 cm gives 141.3 cm² using π = 3.14.

Add one large semicircle and two smaller semicircular arcs. The large diameter equals the sum of the two small diameters.

Use (n - 2) × 180° for the sum of interior angles. Find one angle of a regular pentagon and a missing hexagon angle.

Use the 180-degree angle sum to solve for x when a triangle's angles are x, 3x, and 5x.

For sides, the two shortest must add to more than the longest. For angles, the sum must be 180°. Then decide whether the measurements force congruent triangles.

Use the extended sine rule to calculate a chord subtended by a 30-degree inscribed angle.

Find the largest riverside plot from a fixed fence length by maximizing a quadratic area function.

Use supplementary angle pairs to find two 45-degree alternate interior angles and prove parallel lines.

To count numbers divisible by both a and b, count multiples of lcm(a,b) in the required range. Worked example for two-digit multiples of 3 and 7.

Use the repeating cycle of last digits instead of calculating the whole power. Worked example: the last digit of 3^144.

Convert 1.151515… to its simplest fraction by shifting the repeated block and subtracting.

Use the least common multiple of the denominators, then multiply each numerator and denominator by the same factor. Worked examples with 5/18 and 7/15, plus mixed numbers.

Learn why terminating and repeating decimals are rational, with classification examples and conversions of mixed numbers to decimal form.

Multiply along each path leading to an event, then add the mutually exclusive path probabilities.

Count each distinct value in a 40-item data set and check that the frequencies sum to 40. Includes a corrected worked example.

Fill in the missing probability so the row adds up to 1, then accumulate the probabilities left to right. Worked example with X = 2, 3, 4 and the step graph of F(x).

The probability of an interval is the difference of the CDF at its ends: P(a < X ≤ b) = F(b) − F(a). Worked example with F(x) = (x³ − 1)/26 and the answer 7/26 ≈ 0.269.

Recover the missing probability, multiply values by probabilities for the mean, then use D(X) = M(X²) − M(X)². Worked example: X = −2, 0, 3 gives M(X) = 0.5 and D(X) = 3.25.

Sort for the median, add and divide for the mean, and count frequencies for the mode. Two worked data sets include an even number of values and three modes.

A card is drawn from 1 through 8, replaced, then drawn again. Multiply the two probabilities to find a joint event.

Use a double number line to find a unit rate, scale ratios, and turn a proportional table into an equation.

The first octagon uses 8 sticks and each new octagon shares one side, so it adds 7. The general formula is 8 + 7(n − 1) = 7n + 1.

Transform a nonlinear-looking recurrence into an arithmetic sequence and find a distant term and threshold.

Pair symmetric sine factors to turn a four-factor trigonometric product into a squared cosine.