Advanced Math Questions.

Explanation

Multiply coefficients: 3 × 4 = 12. Add exponents: x² · x³ = x⁵. Result: 12x⁵.

Explanation

FOIL: x² − 3x + 4x − 12 = x² + x − 12.

Explanation

Difference of squares: a² − b² = (a + b)(a − b). So x² − 9 = (x + 3)(x − 3).

Explanation

Factor the numerator: (x + 3)(x + 2). Cancel (x + 2): result is x + 3.

Explanation

(2x − 3)² = 4x² − 12x + 9. Check: (2x)² = 4x², 2(2x)(−3) = −12x, (−3)² = 9. ✓

Explanation

Divide each term: 2x³/2x − 6x/2x = x² − 3.

Explanation

Combine like terms: (3+1)x² + (2−5)x + (−1+4) = 4x² − 3x + 3.

Explanation

(x + 2)² = x² + 2(x)(2) + 4 = x² + 4x + 4.

Explanation

Factor: (x − 2)(x − 3) = 0. So x = 2 or x = 3.

Explanation

Factor: (x + 5)(x − 1) = 0. So x = −5 or x = 1.

Explanation

Discriminant D = 36 − 36 = 0. One repeated solution: x = 3.

Explanation

D = b² − 64 = 0 → b² = 64 → b = ±8. Both 8 and −8 work; 8 is listed.

Explanation

h = −(−4)/(2·1) = 2. k = 4 − 8 + 7 = 3. Vertex: (2, 3).

Explanation

D = 36 − 4c < 0 → c > 9. The only choice greater than 9 is c = 10.

Explanation

2x² = 8 → x² = 4 → x = ±2.

Explanation

(x − 3)(x + 1) = x² + x − 3x − 3 = x² − 2x − 3 = 0.

Explanation

Square both sides: x + 5 = 16 → x = 11.

Explanation

Cross-multiply: 5(x − 2) = 3(x + 1) → 5x − 10 = 3x + 3 → 2x = 13 → x = 6.5.

Explanation

x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0. Two solutions: x = 2 and x = −1.

Explanation

x² + 1 = 2x + 1 → x² − 2x = 0 → x(x − 2) = 0 → x = 0 or x = 2.

Explanation

x² − 4 = −x + 2 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0. x-values: −3 and 2. Sum = −1.

Explanation

x² = 4 → x = ±2.

Explanation

x² + 3x = kx → x(x + 3 − k) = 0. For x = 4: 4 + 3 − k = 0 → k = 7.

Explanation

D = 4 + 12 = 16 > 0, so two x-intercepts. Factor: (x − 3)(x + 1) = 0 → x = 3 and x = −1.

Explanation

f(3) = 3² + 2 = 9 + 2 = 11.

Explanation

f(0) = 2 · (1.5)⁰ = 2 · 1 = 2.

Explanation

Vertex form with vertex (h, k) = (3, 5) and opening downward (negative a): y = −(x − 3)² + 5.

Explanation

Vertex x = 12/(2·3) = 2. f(2) = 12 − 24 + 7 = −5.

Explanation

2⁵ = 32.

Explanation

f(0) = a · 1 = a = 4. Verify: f(3) = 4 · 8 = 32. ✓

Explanation

Vertex x = −6/(2 · −1) = 3. f(3) = −9 + 18 − 5 = 4.

Explanation

g(2) = 2(2) + 1 = 5. f(5) = 25 − 4 = 21.

Explanation

f(4) = 3(4) − 2 = 12 − 2 = 10.

Explanation

g(−1) = (−1)² − 3(−1) + 2 = 1 + 3 + 2 = 6.

Explanation

f(2) = 5. f(f(2)) = f(5) = 2(5) + 1 = 11.

Explanation

Shifting left by 2 replaces x with (x + 2); shifting up by 3 adds 3. Result: y = f(x + 2) + 3.

Explanation

h(3) = f(3) + 5 = 7 + 5 = 12.

Explanation

f(−x) = −x³ + x = −f(x). So f(−x) = f(x) only when f(x) = 0: x³ − x = 0 → x(x−1)(x+1) = 0 → x = 0, ±1.

Explanation

f(−1) = |−1 − 3| = |−4| = 4.

Explanation

Replacing x with (x − 1) shifts the graph right by 1. Multiplying by 2 stretches it vertically by a factor of 2.

Explanation

Multiply coefficients: 5 × (−2) = −10. Add exponents: x² · x⁴ = x⁶. Result: −10x⁶.

Explanation

Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x + 3)(x + 4).

Explanation

f(−3) = (−3)² − 1 = 9 − 1 = 8.

Explanation

x² − 3x − 18 = 0 → (x − 6)(x + 3) = 0 → x = 6 or x = −3.

Explanation

Difference of squares: (a + b)(a − b) = a² − b². (2x)² − 3² = 4x² − 9.

Explanation

In vertex form y = (x − h)² + k, the vertex is (h, k). Here h = 2 and k = 5, so vertex is (2, 5).

Explanation

3x² = 27 → x² = 9 → x = ±3.

Explanation

p(2) = 2³ − 4(2) = 8 − 8 = 0.

Explanation

Factor the numerator: (x − 2)². Divide by (x − 2): result is x − 2.

Explanation

f(4) = 3 · 2⁴ = 3 · 16 = 48.

Explanation

D = b² − 4ac = 25 − 4(2)(3) = 25 − 24 = 1 > 0. Two distinct real solutions.

Explanation

By Vieta's formulas, sum of roots = −b/a = −(−8)/2 = 4.

Explanation

The parabola opens downward (a = −2 < 0). Vertex is at (−1, 8), so maximum value is 8, occurring at x = −1.

Explanation

Set equal: x² − 2x + 3 = x + 1 → x² − 3x + 2 = 0. D = 9 − 8 = 1 > 0, so two distinct real solutions: (x − 1)(x − 2) = 0 → x = 1 or x = 2.

Explanation

(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 = 0.

Explanation

Numerator: x³ · x² = x⁵. Then x⁵/x⁴ = x¹ = x.

Explanation

(3x − 2)² = 9x² − 2(3x)(2) + 4 = 9x² − 12x + 4.

Explanation

g(3) = 5 · (0.5)³ = 5 · 0.125 = 0.625.

Explanation

f(x) = a(x + 1)(x − 4). At x = 0: f(0) = a(1)(−4) = −4a = −8 → a = 2. The leading coefficient is 2.

Explanation

By Vieta's formulas, product of roots = c/a = 10/1 = 10.

Explanation

f(2) = 4 + 4 = 8. g(8) = 8 − 3 = 5.

Explanation

D = 16 − 16 = 0. One repeated solution. (2x + 1)² = 0 → x = −1/2.

Explanation

Reflecting across the x-axis gives y = −f(x). Shifting up 4 adds 4: y = −f(x) + 4.

Explanation

Square both sides: 2x − 3 = 25 → 2x = 28 → x = 14.

Explanation

Vertex form: y = a(x − 1)² − 4. At (3, 4): 4 = a(4) − 4 → 4a = 8 → a = 2.

Explanation

Using the product rule: 2³ × 2² = 2^(3+2) = 2⁵ = 32.

Explanation

f(−2) = (−2)² + 3(−2) − 4 = 4 − 6 − 4 = −6.

Explanation

The expression is undefined when the denominator equals zero: x − 3 = 0, so x = 3.

Explanation

Factor: (x + 5)(x − 1) = 0. So x = −5 or x = 1.

Explanation

First, g(3) = 3² = 9. Then f(9) = 2(9) + 1 = 19.

Explanation

Using the product rule of exponents: x⁶ · x⁻² = x^(6+(−2)) = x⁴.

Explanation

g(0) = 0² − 2(0) + 1 = 0 − 0 + 1 = 1.

Explanation

√48 = √(16 × 3) = √16 · √3 = 4√3.

Explanation

Factor: x² + 7x + 12 = (x + 3)(x + 4). So (x + 3) is a factor.

Explanation

In vertex form y = (x − h)² + k, the vertex is (h, k). Here h = 3 and k = 5, so the vertex is (3, 5).

Explanation

2x² − 8 = 0 → 2x² = 8 → x² = 4 → x = ±2.

Explanation

Distribute the negative: 3x² + 2x − 1 − x² + 4x − 5 = 2x² + 6x − 6.

Explanation

Since x² ≥ 0 for all real x, f(x) = x² + 3 ≥ 0 + 3 = 3. The minimum value is 3, achieved at x = 0.

Explanation

Multiply by adding exponents: x^(3+2) · y^(2+4) = x⁵y⁶.

Explanation

Discriminant = b² − 4ac = (−4)² − 4(2)(2) = 16 − 16 = 0. A discriminant of 0 means exactly one real solution.

Explanation

By Vieta's formulas, for x² + bx + c = 0 the sum of roots = −b. Here the sum of roots = −(−6) = 6.

Explanation

Factor: (x − 4)(x + 3) > 0. The expression is positive when both factors are positive (x > 4) or both negative (x < −3).

Explanation

(f ∘ g)(3) = f(g(3)). First g(3) = 2(3) − 1 = 5. Then f(5) = 5² = 25.

Explanation

For x² + 2x + 5 = 0, the discriminant = 2² − 4(1)(5) = 4 − 20 = −16 < 0, so there are no real solutions.

Explanation

64 = 4³, so 4^x = 4³ → x = 3.

Explanation

The population doubles every 3 hours, so after t hours it has doubled t/3 times: P(t) = 500 · 2^(t/3).

Explanation

Complete the square: f(x) = 2(x² − 4x) + 9 = 2(x − 2)² − 8 + 9 = 2(x − 2)² + 1. The minimum is 1 at x = 2.

Explanation

Factor the numerator: (x² − 9) = (x + 3)(x − 3). Cancel (x + 3): the expression simplifies to x − 3.

Explanation

log₂(x) = 5 means 2⁵ = x. So x = 32.

Explanation

Set f(x) = 0: x³ − 4x = x(x² − 4) = x(x − 2)(x + 2) = 0. The solutions are x = 0, 2, −2 — three x-intercepts.