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Distributing the 3 gives 6x - 12, and combining like terms with 5x yields 6x + 5x - 12 = 11x - 12. Like terms are combined by adding their coefficients while keeping the variable part unchanged.
Subtract 3x from both sides to get 2x - 7 = 9, then add 7 to both sides to get 2x = 16, and divide by 2 to obtain x = 8. Each step must be applied to both sides to preserve equality.
Slope equals the change in y divided by the change in x: (11 - 3)/(6 - 2) = 8/4 = 2. The order of subtraction must be consistent between numerator and denominator.
Substituting x = -2 gives 2(4) - 3(-2) + 1 = 8 + 6 + 1 = 15. Squaring a negative yields a positive, and multiplying two negatives yields a positive.
The quadratic factors as (x - 2)(x - 3) because -2 and -3 multiply to 6 and add to -5. A factor is one of these binomials.
A rational function is undefined where its denominator equals zero, so x - 4 = 0 gives x = 4. That single value is excluded from the domain.
The y-intercept occurs where x = 0, giving 3y = 12 and y = 4, so the point is (0, 4). An intercept is written as a coordinate pair with the other variable set to zero.
An arithmetic sequence has a constant common difference between consecutive terms; here each term increases by 3. The other lists change by varying amounts or by a constant ratio.
Subtract 5 to get -2x > 6, then divide by -2 and reverse the inequality sign to get x < -3. Dividing or multiplying an inequality by a negative number flips its direction.
First evaluate the inner function: g(2) = 2(2) - 3 = 1. Then apply f to that result: f(1) = 1 + 1 = 2, so f(g(2)) = 2.
The vertex form y = (x - h)^2 + k has vertex (h, k). Here h = 3 and k = -5, giving (3, -5).
When multiplying powers with the same base, add the exponents: x^3 times x^4 = x^(3+4) = x^7. The base stays the same and the coefficients (both 1) multiply to 1.
Using point-slope form, y - 2 = 3(x - 1), which expands to y = 3x - 3 + 2 = 3x - 1. The y-intercept must satisfy the given point.
A rational expression is undefined when its denominator is zero. Setting x + 2 = 0 gives x = -2, the excluded value.
Substitute a + 1: (a + 1)^2 + 2(a + 1) = a^2 + 2a + 1 + 2a + 2 = a^2 + 4a + 3. Expanding the square and distributing the 2 must both be done before combining terms.
Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x - y = 2 gives y = 1, so the solution is (3, 1), which satisfies both equations.
Write y = 4x - 8, swap x and y to get x = 4y - 8, then solve for y: y = (x + 8)/4. The inverse undoes each operation in reverse order, so the -8 becomes +8 before dividing by 4.
Factor as (x + 1)(x + 5) = 0, giving x = -1 and x = -5. The roots are the values that make each factor zero, so their signs are opposite to those in the factors.
In y = (x - h)^2 + k, the graph shifts horizontally by h and vertically by k. Here h = 2 shifts right and k = 3 shifts up.
The first term is 5 and the common difference is 4, so the 20th term is 5 + 19(4) = 81. The sum formula S = n(a1 + an)/2 gives 20(5 + 81)/2 = 10(86) = 860... wait, recomputing: 20(86)/2 = 860.
A logarithm answers the question 'to what power must the base be raised?' Since 2^5 = 32, log base 2 of 32 is 5.
Set 3a + 2 = 17 and solve: subtract 2 to get 3a = 15, then divide by 3 to get a = 5. This reverses the function rule to recover the input.
The numerator is a difference of squares: x^2 - 9 = (x - 3)(x + 3). Cancelling the common factor (x + 3) leaves x - 3, valid wherever x + 3 is nonzero.
Because the coefficient -2 is negative, the parabola opens downward and its vertex is a maximum. Vertex form gives the vertex at x = -1, where f(-1) = 8.
Since f(x + 1) = 3 * 2^(x+1) = 2 * (3 * 2^x) = 2 f(x), each unit increase in x multiplies the output by 2. The base of the exponential controls this multiplicative growth factor.
The slope of a line through two points (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1). Substituting the given points gives (13 - 5)/(6 - 2) = 8/4 = 2. Thus the slope is 2.
Parallel lines have the same slope. The given line y = 2x + 1 has slope 2, so any line with slope 2 is parallel. The equation y = 2x - 3 also has slope 2 and is therefore parallel.
To evaluate f(3), substitute x = 3 into the expression: 2(3)^2 - 5 = 2(9) - 5 = 18 - 5 = 13. Thus f(3) = 13.
The equation |x - 3| = 5 means that the distance between x and 3 is 5. This gives two cases: x - 3 = 5 or x - 3 = -5. Solving yields x = 8 or x = -2, so the solution set is {8, -2}.
We need two numbers that multiply to 6 and add to -5. The numbers -2 and -3 satisfy this, so x^2 - 5x + 6 = (x - 2)(x - 3). Expanding confirms the product.
When raising a product to a power, apply the exponent to each factor: (2x^3y^2)^3 = 2^3 * (x^3)^3 * (y^2)^3. This equals 8x^9y^6 because 2^3 = 8 and powers are multiplied.
To find the inverse, set y = 3x - 1 and solve for x: y + 1 = 3x, so x = (y + 1)/3. Interchanging x and y gives f^{-1}(x) = (x + 1)/3.
The logarithm log_3 81 asks for the exponent to which 3 must be raised to obtain 81. Since 3^4 = 81, the value is 4.
Adding the two equations eliminates y: (x + y) + (x - y) = 10 + 2, so 2x = 12 and x = 6. Substituting x = 6 into x + y = 10 gives y = 4.
The expression under a square root must be nonnegative, so 2x - 6 ≥ 0. Solving gives 2x ≥ 6, hence x ≥ 3. Therefore the domain is all real numbers greater than or equal to 3.
The equation is in vertex form y = a(x - h)^2 + k, where the vertex is (h, k). Here h = 1 and k = 3, so the vertex is (1, 3). The axis of symmetry is the vertical line x = h, which is x = 1.
The first term is a = 3 and the common ratio is r = 2. The sum of the first n terms of a geometric sequence is S_n = a(1 - r^n)/(1 - r). For n = 10, S_10 = 3(1 - 2^10)/(1 - 2) = 3(1 - 1024)/(-1) = 3(1023) = 3069.
Factor the numerator as (x - 2)(x + 2) and the denominator as (x - 2)(x + 1). Canceling the common factor (x - 2) gives (x + 2)/(x + 1).
Write 8 as a power of 2: 8 = 2^3. Then the equation becomes 2^(x+1) = 2^3. Since the bases are equal, the exponents must be equal: x + 1 = 3, so x = 2.
The composition (f ∘ g)(x) means f(g(x)). Substitute g(x) = x - 2 into f: f(x - 2) = (x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5.
Factor the quadratic: x^2 - 4x + 3 = (x - 1)(x - 3). The critical points are x = 1 and x = 3. Testing intervals shows the product is positive when x < 1 or x > 3, so the solution is x < 1 or x > 3.
By the Remainder Theorem, the remainder is P(2). Compute P(2) = 2^3 - 2(2^2) + 4(2) - 5 = 8 - 8 + 8 - 5 = 3. Thus the remainder is 3.
Find the derivative: f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1). Critical points are x = 2 and x = -1. The second derivative is f''(x) = 12x - 6. At x = -1, f''(-1) = -18 < 0, indicating a local maximum.
The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Substituting h = 2, k = -3, and r = 4 gives (x - 2)^2 + (y + 3)^2 = 16.
Write both sides with base 3: 27 = 3^3, so 27^(x - 1) = (3^3)^(x - 1) = 3^(3x - 3). The equation becomes 3^(2x) = 3^(3x - 3). Equating exponents gives 2x = 3x - 3, so x = 3.
Factor by grouping or using the factor theorem: f(1) = 1 - 3 + 2 = 0, so x - 1 is a factor. Dividing gives x^2 + x - 2 = (x - 1)(x + 2). Thus f(x) = (x - 1)^2(x + 2), showing a double root at x = 1.
Set y = (x + 1)/(x - 2) and solve for x: y(x - 2) = x + 1, so yx - 2y = x + 1. Rearranging gives x(y - 1) = 2y + 1, hence x = (2y + 1)/(y - 1). Interchanging x and y gives f^{-1}(x) = (2x + 1)/(x - 1).
First find f^{-1}(x): y = 2x + 3, so x = (y - 3)/2. Thus f^{-1}(5) = (5 - 3)/2 = 1. Then g(f^{-1}(5)) = g(1) = 1^2 = 1? Wait, that gives 1, not 4. Let's recompute: f^{-1}(5) = (5-3)/2 = 1, g(1)=1. So correct answer should be 1, but options are 4,16,13,25. I need to adjust the problem to make correct answer 4. Change to f(x)=2x+1? Then f^{-1}(5)=(5-1)/2=2, g(2)=4. So change f(x) to 2x+1. Let's do that: f(x)=2x+1, g(x)=x^2. Then (g∘f^{-1})(5)=g(2)=4. So correct option 4. Options: 4,16,13,25. Explanation: find f^{-1}(5)=2, then g(2)=4. So correctIndex 0. I'll set question accordingly.
Use synthetic division by 2: coefficients 1, -2, -3, 8, -4 give quotient x^3 - 3x + 2 with remainder 0. Divide again by 2: coefficients 1, 0, -3, 2 give quotient x^2 + 2x - 1 with remainder 0. Divide a third time by 2: coefficients 1, 2, -1 give quotient x + 4 with remainder 7, not 0. So the factor (x - 2) appears twice, meaning the multiplicity is 2.
Combine the logarithms: log_2(x(x - 2)) = 3, so x(x - 2) = 2^3 = 8. This gives x^2 - 2x - 8 = 0, which factors as (x - 4)(x + 2) = 0. Thus x = 4 or x = -2. Since the domain of log requires x > 0 and x - 2 > 0, x = -2 is extraneous, leaving x = 4.