#1
Solve the equation: 2x - 7 = 3x + 5
x = -6
ExplanationIsolate x by moving terms around and combining like terms.
#2
Expand and simplify the expression: (a - 2)(3a + 4)
3a^2 + 4a - 8
ExplanationApply the distributive property and combine like terms.
#3
Simplify the expression: (2a^2b^3)^2 / (4ab)^2
4a^2b^2
ExplanationApply the power rule and simplify the expression.
#4
Find the value of x in the equation: 2^(3x - 1) = 8
x = 1
ExplanationUse logarithms to solve for x.
#5
Solve the inequality: 2x - 5 > 3x + 7
x < -12
ExplanationIsolate x and determine the direction of the inequality.
#6
Expand and simplify the expression: (2x - 1)(x^2 + x + 1)
2x^3 + x^2 + 2x - 1
ExplanationApply the distributive property and combine like terms.
#7
Simplify the expression: 3x + 2(x - 5) = 4(2x + 1)
x = -3
ExplanationCombine like terms and solve for x.
#8
Factorize the quadratic expression: x^2 - 5x + 6
(x - 2)(x - 3)
ExplanationFind two numbers that multiply to the constant term and add up to the coefficient of the linear term.
#9
Solve the system of equations:
2x + y = 8
3x - 2y = 1
x = 3, y = 2
ExplanationUse elimination or substitution to solve for x and y.
#10
Factorize the following expression: 16x^2 - 9y^2
(4x + 3y)(4x - 3y)
ExplanationApply the difference of squares formula.
#11
Solve for x: log₂(x + 4) = 3
x = 8
ExplanationUse the definition of logarithms to solve for x.
#12
Simplify the expression: (a^2 - b^2)/(a + b)
a - b
ExplanationFactor the numerator and cancel common factors.
#13
If f(x) = 2x^3 - 5x^2 + 3x + 7, find f'(x) (derivative of f(x)).
f'(x) = 6x^2 - 10x + 3
ExplanationApply the power rule to find the derivative of each term.
#14
If g(x) = (x^2 + 2x + 1)^3, find g'(x) (derivative of g(x)).
g'(x) = 3(x^2 + 2x + 1)^2(2x + 2)
ExplanationApply the chain rule to find the derivative of the composite function.
#15
If f(x) = 4x³ - 6x² + 2x + 1, find f''(x) (second derivative of f(x)).
f''(x) = 12x - 6
ExplanationFind the first derivative and then differentiate again.
#16
If h(x) = (2x^2 + 3x - 1)^4, find h'(x) (derivative of h(x)).
h'(x) = 4(2x^2 + 3x - 1)^3(4x + 3)
ExplanationApply the chain rule to find the derivative of the composite function.
#17
If k(x) = (3x^2 - 4x + 1)^5, find k'(x) (derivative of k(x)).
k'(x) = 5(3x^2 - 4x + 1)^4(3x - 2)
ExplanationApply the chain rule to find the derivative of the composite function.