Algebra, Geometry, Trigonometry, And Calculus Quiz

To factor x² − 2x − 24, identify two numbers whose product equals −24 and sum equals −2. The numbers −6 and 4 satisfy both conditions. Rewriting the middle term as −6x + 4x allows grouping. Factoring each group produces (x − 6)(x + 4). This verifies the factorization because expanding the product returns the original quadratic expression exactly without remainder or mismatch.

Factoring 6x² + 17x + 12 requires identifying binomials whose product equals the quadratic. Testing (3x + 4)(2x + 3) produces 6x² + 9x + 8x + 12. Combining middle terms gives 17x, matching the expression. Since coefficients and constants align perfectly, this confirms the correct factorization without excess or missing terms, validating the binomial pairing analytically.

Solving the system involves substitution or elimination. From x + 2y = 9, isolate x as 9 − 2y. Substituting into 3x − 4y = −33 gives 27 − 6y − 4y = −33. Simplifying yields −10y = −60, so y equals 6. Substituting back gives x equals −3, satisfying both equations accurately.

The slope between points (3,4) and (1,−10) is calculated as (4 − (−10)) divided by (3 − 1), resulting in 14/2 or 7. Using y = mx + b and substituting point (3,4) gives 4 = 21 + b, so b equals −17. The resulting equation y = 7x − 17 correctly represents the line through both points.

A regular polygon’s exterior angles always sum to 360 degrees. A heptagon has seven sides, so each exterior angle equals 360 divided by 7. Performing the calculation gives approximately 51.4 degrees. This applies only to regular polygons where all sides and angles are equal, ensuring uniform exterior angles around the shape without variation or distortion.

Similar triangles have proportional corresponding sides. Setting the ratio AB/DE equal to BC/EF yields 16/DE = 5/7. Cross-multiplying gives 5DE = 112, leading to DE = 22.4. This proportional reasoning ensures scale consistency between triangles and confirms the calculated length accurately matches similarity rules without approximation errors.

The radius of the sphere equals half the diameter, giving r = 2.5 meters. Substituting into V = (4/3)πr³ yields (4/3)π(15.625). Multiplying produces approximately 20.8π cubic meters. This calculation follows standard geometric volume formulas and confirms correct unit conversion and exponent application throughout the process.

The slope of a tangent line equals the derivative evaluated at that point. Differentiating y = x³ gives y′ = 3x². Substituting x = 2 results in 3(4) = 12. This value represents the instantaneous rate of change at x = 2 and defines the tangent line’s slope precisely at that coordinate.

Applying the chain rule, the derivative of sin(u) equals cos(u) multiplied by du/dx. Here, u = 2x + 5, whose derivative is 2. Therefore, the derivative becomes 2cos(2x + 5). This accounts for both the outer trigonometric function and the inner linear function accurately and completely.

Expanding (x + 4)² gives x² + 8x + 16. Integrating term by term yields x³/3 + 4x² + 16x + C. This method applies the power rule correctly to each term, ensuring constants and exponents are handled systematically and the resulting antiderivative matches the original integrand when differentiated.