没有符合这些筛选条件的题目。
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Which expression is equivalent to (3x²)(4x³)?
- 7x⁵
- 12x⁵ ✓
- 12x⁶
- 7x⁶
解析
Multiply coefficients: 3 × 4 = 12. Add exponents: x² · x³ = x⁵. Result: 12x⁵.
-
Which is equivalent to (x + 4)(x − 3)?
- x² + x + 12
- x² − x − 12
- x² + x − 12 ✓
- x² − 7x − 12
解析
FOIL: x² − 3x + 4x − 12 = x² + x − 12.
-
Which shows the complete factorization of x² − 9?
- (x + 3)²
- (x − 3)²
- (x + 3)(x − 3) ✓
- x(x − 9)
解析
Difference of squares: a² − b² = (a + b)(a − b). So x² − 9 = (x + 3)(x − 3).
-
Which expression is equivalent to (x² + 5x + 6)/(x + 2), where x ≠ −2?
- x + 2
- x + 3 ✓
- x − 3
- x² + 3x
解析
Factor the numerator: (x + 3)(x + 2). Cancel (x + 2): result is x + 3.
-
Which expression is equivalent to 4x² − 12x + 9?
- (2x + 3)²
- (2x − 3)² ✓
- (4x − 3)(x − 3)
- (4x + 3)(x − 3)
解析
(2x − 3)² = 4x² − 12x + 9. Check: (2x)² = 4x², 2(2x)(−3) = −12x, (−3)² = 9. ✓
-
Which is equivalent to (2x³ − 6x)/(2x), where x ≠ 0?
- x² − 6
- 2x² − 3
- x² − 3 ✓
- x³ − 3
解析
Divide each term: 2x³/2x − 6x/2x = x² − 3.
-
What is (3x² + 2x − 1) + (x² − 5x + 4)?
- 4x² + 7x + 3
- 4x² − 3x + 3 ✓
- 4x² − 3x − 3
- 2x² − 3x + 3
解析
Combine like terms: (3+1)x² + (2−5)x + (−1+4) = 4x² − 3x + 3.
-
Which is equivalent to (x + 2)²?
- x² + 4
- x² + 2x + 4
- x² + 4x + 4 ✓
- x² − 4x + 4
解析
(x + 2)² = x² + 2(x)(2) + 4 = x² + 4x + 4.
-
What are the solutions to x² − 5x + 6 = 0?
- x = 1 and x = 6
- x = 2 and x = 3 ✓
- x = −2 and x = −3
- x = 2 and x = −3
解析
Factor: (x − 2)(x − 3) = 0. So x = 2 or x = 3.
-
What are the solutions to x² + 4x − 5 = 0?
- x = 1 and x = −5 ✓
- x = −1 and x = 5
- x = 5 and x = −1
- x = 2 and x = −5
解析
Factor: (x + 5)(x − 1) = 0. So x = −5 or x = 1.
-
For the equation x² − 6x + 9 = 0, how many distinct real solutions are there?
- 0
- 1 ✓
- 2
- 3
解析
Discriminant D = 36 − 36 = 0. One repeated solution: x = 3.
-
The equation x² + bx + 16 = 0 has exactly one real solution. What is a possible value of b?
- −4
- 4
- 8 ✓
- 16
解析
D = b² − 64 = 0 → b² = 64 → b = ±8. Both 8 and −8 work; 8 is listed.
-
What is the vertex of y = x² − 4x + 7?
- (2, 3) ✓
- (−2, 3)
- (2, −3)
- (4, 7)
解析
h = −(−4)/(2·1) = 2. k = 4 − 8 + 7 = 3. Vertex: (2, 3).
-
For what value of c does x² + 6x + c = 0 have no real solutions?
- c = 8
- c = 9
- c = 10 ✓
- c = 36
解析
D = 36 − 4c < 0 → c > 9. The only choice greater than 9 is c = 10.
-
What are the solutions to 2x² − 8 = 0?
- x = ±1
- x = ±2 ✓
- x = ±4
- x = 2 only
解析
2x² = 8 → x² = 4 → x = ±2.
-
A quadratic equation has solutions x = 3 and x = −1. Which could be the equation?
- x² + 2x − 3 = 0
- x² − 2x − 3 = 0 ✓
- x² − 2x + 3 = 0
- x² + 2x + 3 = 0
解析
(x − 3)(x + 1) = x² + x − 3x − 3 = x² − 2x − 3 = 0.
-
What is the value of x if √(x + 5) = 4?
- 3
- 9
- 11 ✓
- 21
解析
Square both sides: x + 5 = 16 → x = 11.
-
What is the positive solution to (x − 2)/(x + 1) = 3/5?
- 5
- 5.5
- 6.5 ✓
- 7
解析
Cross-multiply: 5(x − 2) = 3(x + 1) → 5x − 10 = 3x + 3 → 2x = 13 → x = 6.5.
-
The system y = x² and y = x + 2 has how many solutions?
- 0
- 1
- 2 ✓
- 3
解析
x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0. Two solutions: x = 2 and x = −1.
-
Where does y = x² + 1 intersect y = 2x + 1? What are the x-values?
- x = 1 and x = 2
- x = 0 and x = 2 ✓
- x = −1 and x = 2
- x = 0 and x = 1
解析
x² + 1 = 2x + 1 → x² − 2x = 0 → x(x − 2) = 0 → x = 0 or x = 2.
-
The parabola y = x² − 4 and the line y = −x + 2 intersect at two points. What is the sum of their x-coordinates?
- −1 ✓
- 1
- 5
- −5
解析
x² − 4 = −x + 2 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0. x-values: −3 and 2. Sum = −1.
-
Where does y = x² intersect y = 4?
- x = 2 only
- x = ±1
- x = ±2 ✓
- x = ±4
解析
x² = 4 → x = ±2.
-
The system y = x² + 3x and y = kx has solutions at x = 0 and x = 4. What is k?
- 4
- 5
- 7 ✓
- 9
解析
x² + 3x = kx → x(x + 3 − k) = 0. For x = 4: 4 + 3 − k = 0 → k = 7.
-
How many x-intercepts does the parabola y = x² − 2x − 3 have?
- 0
- 1
- 2 ✓
- 3
解析
D = 4 + 12 = 16 > 0, so two x-intercepts. Factor: (x − 3)(x + 1) = 0 → x = 3 and x = −1.
-
If f(x) = x² + 2, what is f(3)?
- 7
- 9
- 11 ✓
- 13
解析
f(3) = 3² + 2 = 9 + 2 = 11.
-
The function f(x) = 2 · (1.5)^x models a population. What is the initial population (at x = 0)?
- 1
- 1.5
- 2 ✓
- 3
解析
f(0) = 2 · (1.5)⁰ = 2 · 1 = 2.
-
A parabola opens downward and has vertex at (3, 5). Which equation could represent it?
- y = (x − 3)² + 5
- y = −(x + 3)² + 5
- y = −(x − 3)² + 5 ✓
- y = (x + 3)² − 5
解析
Vertex form with vertex (h, k) = (3, 5) and opening downward (negative a): y = −(x − 3)² + 5.
-
What is the minimum value of f(x) = 3x² − 12x + 7?
- −7
- −5 ✓
- 3
- 7
解析
Vertex x = 12/(2·3) = 2. f(2) = 12 − 24 + 7 = −5.
-
If f(x) = 2^x, what is f(5)?
- 10
- 16
- 25
- 32 ✓
解析
2⁵ = 32.
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The function f(x) = a · 2^x passes through (0, 4) and (3, 32). What is a?
- 2
- 3
- 4 ✓
- 8
解析
f(0) = a · 1 = a = 4. Verify: f(3) = 4 · 8 = 32. ✓
-
For f(x) = −x² + 6x − 5, what is the maximum value?
- 3
- 4 ✓
- 5
- 6
解析
Vertex x = −6/(2 · −1) = 3. f(3) = −9 + 18 − 5 = 4.
-
If f(x) = x² − 4 and g(x) = 2x + 1, what is f(g(2))?
- 17
- 19
- 21 ✓
- 25
解析
g(2) = 2(2) + 1 = 5. f(5) = 25 − 4 = 21.
-
If f(x) = 3x − 2, what is f(4)?
- 8
- 9
- 10 ✓
- 11
解析
f(4) = 3(4) − 2 = 12 − 2 = 10.
-
If g(x) = x² − 3x + 2, what is g(−1)?
- 4
- 5
- 6 ✓
- 7
解析
g(−1) = (−1)² − 3(−1) + 2 = 1 + 3 + 2 = 6.
-
If f(x) = 2x + 1, what is f(f(2))?
- 9
- 10
- 11 ✓
- 12
解析
f(2) = 5. f(f(2)) = f(5) = 2(5) + 1 = 11.
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The graph of y = f(x) is shifted 3 units up and 2 units to the left. Which equation represents the new function?
- y = f(x − 2) + 3
- y = f(x + 2) + 3 ✓
- y = f(x − 2) − 3
- y = f(x + 3) + 2
解析
Shifting left by 2 replaces x with (x + 2); shifting up by 3 adds 3. Result: y = f(x + 2) + 3.
-
If h(x) = f(x) + 5 and f(3) = 7, what is h(3)?
- 10
- 11
- 12 ✓
- 14
解析
h(3) = f(3) + 5 = 7 + 5 = 12.
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For f(x) = x³ − x, the equation f(−x) = f(x) holds for which values of x?
- x = 0 only
- x = 0 and x = 1
- x = 0, x = 1, and x = −1 ✓
- x = ±1 only
解析
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.
-
If f(x) = |x − 3|, what is f(−1)?
- 2
- 3
- 4 ✓
- 5
解析
f(−1) = |−1 − 3| = |−4| = 4.
-
The function g(x) = 2f(x − 1) is a transformation of f(x). Which transformations are applied?
- Shift left 1 and vertical stretch by 2
- Shift right 1 and vertical stretch by 2 ✓
- Shift right 1 and vertical shrink by 2
- Shift left 1 and vertical shrink by 2
解析
Replacing x with (x − 1) shifts the graph right by 1. Multiplying by 2 stretches it vertically by a factor of 2.
-
Which is equivalent to (5x²)(−2x⁴)?
- −10x⁶ ✓
- −10x⁸
- 3x⁶
- 10x⁶
解析
Multiply coefficients: 5 × (−2) = −10. Add exponents: x² · x⁴ = x⁶. Result: −10x⁶.
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What is the fully factored form of x² + 7x + 12?
- (x + 2)(x + 6)
- (x + 3)(x + 4) ✓
- (x + 1)(x + 12)
- (x − 3)(x − 4)
解析
Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x + 3)(x + 4).
-
If f(x) = x² − 1, what is f(−3)?
- 6
- 7
- 8 ✓
- 9
解析
f(−3) = (−3)² − 1 = 9 − 1 = 8.
-
What is the solution to x² = 3x + 18?
- x = 6 and x = −3 ✓
- x = −6 and x = 3
- x = 6 and x = 3
- x = −6 and x = −3
解析
x² − 3x − 18 = 0 → (x − 6)(x + 3) = 0 → x = 6 or x = −3.
-
Which expression is equivalent to (2x + 3)(2x − 3)?
- 4x² − 9 ✓
- 4x² + 9
- 4x² − 6x + 9
- 4x² + 12x + 9
解析
Difference of squares: (a + b)(a − b) = a² − b². (2x)² − 3² = 4x² − 9.
-
The graph of y = (x − 2)² + 5 has its vertex at which point?
- (−2, 5)
- (2, 5) ✓
- (2, −5)
- (5, 2)
解析
In vertex form y = (x − h)² + k, the vertex is (h, k). Here h = 2 and k = 5, so vertex is (2, 5).
-
What are the solutions to 3x² − 27 = 0?
- x = ±1
- x = ±3 ✓
- x = ±9
- x = 3 only
解析
3x² = 27 → x² = 9 → x = ±3.
-
The polynomial p(x) = x³ − 4x is evaluated at x = 2. What is p(2)?
- 0 ✓
- 2
- 4
- 8
解析
p(2) = 2³ − 4(2) = 8 − 8 = 0.
-
Which expression equals (x² − 4x + 4)/(x − 2) for x ≠ 2?
- x + 2
- x − 2 ✓
- x² − 2
- x + 4
解析
Factor the numerator: (x − 2)². Divide by (x − 2): result is x − 2.
-
An exponential function f(x) = 3 · 2^x. What is f(4)?
- 24
- 36
- 48 ✓
- 12
解析
f(4) = 3 · 2⁴ = 3 · 16 = 48.
-
What is the discriminant of 2x² − 5x + 3 = 0, and how many real solutions does it have?
- D = 1; two real solutions ✓
- D = 25; two real solutions
- D = −1; no real solutions
- D = 0; one real solution
解析
D = b² − 4ac = 25 − 4(2)(3) = 25 − 24 = 1 > 0. Two distinct real solutions.
-
What is the sum of the roots of 2x² − 8x + 6 = 0?
- 2
- 3
- 4 ✓
- 6
解析
By Vieta's formulas, sum of roots = −b/a = −(−8)/2 = 4.
-
For f(x) = −2(x + 1)² + 8, what is the maximum value and where does it occur?
- Maximum of 8 at x = −1 ✓
- Maximum of 8 at x = 1
- Maximum of −1 at x = 8
- Maximum of 6 at x = 0
解析
The parabola opens downward (a = −2 < 0). Vertex is at (−1, 8), so maximum value is 8, occurring at x = −1.
-
How many solutions does the system y = x² − 2x + 3 and y = x + 1 have?
- 0
- 1
- 2 ✓
- 3
解析
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.
-
Which equation has roots x = 5 and x = −2?
- x² − 3x − 10 = 0 ✓
- x² + 3x − 10 = 0
- x² − 3x + 10 = 0
- x² + 7x − 10 = 0
解析
(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 = 0.
-
Simplify: (x³ · x²)/x⁴
- x ✓
- x²
- x³
- x⁴
解析
Numerator: x³ · x² = x⁵. Then x⁵/x⁴ = x¹ = x.
-
Which expression is equivalent to (3x − 2)² ?
- 9x² − 4
- 9x² + 4
- 9x² − 12x + 4 ✓
- 9x² + 12x + 4
解析
(3x − 2)² = 9x² − 2(3x)(2) + 4 = 9x² − 12x + 4.
-
The function g(x) = 5 · (0.5)^x. What is g(3)?
- 0.5
- 0.625 ✓
- 1.25
- 2.5
解析
g(3) = 5 · (0.5)³ = 5 · 0.125 = 0.625.
-
A quadratic function has x-intercepts at x = −1 and x = 4, and passes through (0, −8). What is the leading coefficient?
- −2
- 2 ✓
- −4
- 4
解析
f(x) = a(x + 1)(x − 4). At x = 0: f(0) = a(1)(−4) = −4a = −8 → a = 2. The leading coefficient is 2.
-
What is the product of the roots of x² − 7x + 10 = 0?
- 7
- 10 ✓
- −7
- −10
解析
By Vieta's formulas, product of roots = c/a = 10/1 = 10.
-
If f(x) = x² + 4 and g(x) = x − 3, what is g(f(2))?
- 3
- 4
- 5 ✓
- 6
解析
f(2) = 4 + 4 = 8. g(8) = 8 − 3 = 5.
-
The equation 4x² + 4x + 1 = 0 has how many distinct real solutions?
- 0
- 1 ✓
- 2
- 4
解析
D = 16 − 16 = 0. One repeated solution. (2x + 1)² = 0 → x = −1/2.
-
The graph of y = f(x) is reflected across the x-axis, then shifted up 4 units. Which equation represents the result?
- y = f(x) + 4
- y = −f(x) − 4
- y = −f(x) + 4 ✓
- y = f(−x) + 4
解析
Reflecting across the x-axis gives y = −f(x). Shifting up 4 adds 4: y = −f(x) + 4.
-
Solve for x: √(2x − 3) = 5.
- 11
- 14 ✓
- 16
- 28
解析
Square both sides: 2x − 3 = 25 → 2x = 28 → x = 14.
-
The parabola y = ax² + bx + c has vertex (1, −4) and passes through (3, 4). What is a?
- 1
- 2 ✓
- 3
- 4
解析
Vertex form: y = a(x − 1)² − 4. At (3, 4): 4 = a(4) − 4 → 4a = 8 → a = 2.