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Distribute the 3 to get 6x − 12, then combine like terms with 5x to obtain 11x − 12. Combining like terms means adding coefficients of the same variable power.
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 in numerator and denominator.
Add 7 to both sides to get 5x = 25, then divide both sides by 5 to get x = 5. Each operation must be applied to both sides to preserve equality.
Squaring −3 gives 9 because a negative times a negative is positive, and cubing −2 gives −8 because an odd power keeps the negative sign. Then 9 + (−8) = 1.
This is a difference of squares, a² − b² = (a − b)(a + b), with a = x and b = 3. The two binomial factors multiply back to x² − 9 because the middle terms cancel.
The discount amount is 0.25 × 40 = 10, so the sale price is 40 − 10 = 30. Equivalently, paying 75% of the original price gives 0.75 × 40 = 30.
Absolute value gives the distance from zero, so |−7| = 7 and |4| = 4. Their sum is 11.
Using y = mx + b with point (1, 2): 2 = 3(1) + b, so b = 2 − 3 = −1. The y-intercept is the value of y when x = 0.
Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting into x − y = 1 gives y = 3.
When multiplying powers with the same base, add exponents: x⁵ · x³ = x⁸. When dividing, subtract exponents: x⁸ / x² = x⁶.
The vertex x-coordinate is −b/(2a) = 6/2 = 3. Substituting x = 3 gives f(3) = 9 − 18 + 5 = −4.
The expression factors as (x − 2)(x − 3) = 0 since −2 and −3 multiply to 6 and add to −5. The zero product property gives x = 2 or x = 3.
Raise every factor to the third power: 2³ = 8, and multiply each exponent by 3 to get x⁶y⁹. The coefficient is cubed, not multiplied by 3.
Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15. Squaring −2 gives positive 4, and −3 times −2 is positive 6.
By the Pythagorean theorem, c² = 6² + 8² = 36 + 64 = 100, so c = 10. This is a 3-4-5 triangle scaled by 2.
Perpendicular slopes are negative reciprocals, so the new slope is −3/2. Using point (4, 1): 1 = (−3/2)(4) + b gives b = 1 + 6 = 7.
The denominator factors as (x − 4)(x + 4). Canceling the common factor (x + 4) leaves 1/(x − 4), valid where x ≠ ±4.
Let the integers be n and n + 1, so 2n + 1 = 47 and n = 23. The larger integer is n + 1 = 24.
Multiply coefficients and radicands: 3·4 = 12 and √2·√8 = √16 = 4. Then 12 · 4 = 48.
The common ratio is 2, and the nth term is 3·2^(n−1). The 7th term is 3·2⁶ = 3·64 = 192.
Subtract 3 to get −2x < 6, then divide by −2 and flip the inequality sign to get x > −3. Dividing by a negative reverses the direction of the inequality.
Combine the logs: log₂[x(x − 2)] = 3, so x² − 2x = 8 and x² − 2x − 8 = 0. This factors as (x − 4)(x + 2) = 0, giving x = 4 or x = −2; only x = 4 keeps both log arguments positive.
The radius is the distance between the center and the point: √[(5 − 2)² + (3 + 1)²] = √(9 + 16) = √25 = 5.
Using the quadratic formula with a = 2, b = 3, c = −2: the discriminant is 9 + 16 = 25, so x = (−3 ± 5)/4. This gives x = 1/2 or x = −2.
In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse (6), and the side opposite 60° is that value times √3. So the side opposite 60° is 6√3.
Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. This ratio measures vertical rise per unit of horizontal run.
Compute 5! = 120, 3! = 6, and 2! = 2, so the expression is 120/(6·2) = 120/12 = 10. This is the binomial coefficient counting ways to choose 2 items from 5.
The midpoint averages the coordinates: ((−3 + 7)/2, (4 + (−2))/2) = (4/2, 2/2) = (2, 1).
Compute each power first: 4³ = 64 and 3² = 9. Then 64 − 9 = 55.
The diagonal forms the hypotenuse of a right triangle with legs 12 and 5, so it equals √(12² + 5²) = √(144 + 25) = √169 = 13.
Factor both numbers: 36 = 2²·3² and 60 = 2²·3·5. The shared factors are 2²·3 = 12, which is the largest divisor of both.
There are 4 red marbles out of 10 total, giving 4/10, which simplifies to 2/5.
Add 7 to both sides to get 3x = 21, then divide by 3 to obtain x = 7.
Solve for y to get slope-intercept form: −3y = −6x + 12, so y = 2x − 4. Parallel lines share the same slope, which is 2.
Factor the numerator as (x − 3)(x + 3) and the denominator as (x + 3)(x + 4). The common factor (x + 3) cancels, leaving (x − 3)/(x + 4).
There are C(4,2) = 6 arrangements with exactly two heads out of 2⁴ = 16 equally likely outcomes, giving 6/16 = 3/8.
Set y = 4x − 8, swap x and y to get x = 4y − 8, then solve for y: y = (x + 8)/4. This undoes the original operations in reverse order.
The first term is 7 and the common difference is 4, so the 20th term is 7 + 19(4) = 83. The sum is (20/2)(7 + 83) = 10·90 = 900.
Factor out 2: 2(x² − 6x) + 7. Complete the square inside: 2((x − 3)² − 9) + 7 = 2(x − 3)² − 18 + 7 = 2(x − 3)² − 11.
A square root requires a nonnegative radicand, so 2x − 10 ≥ 0, which gives x ≥ 5. The endpoint x = 5 is included because the radicand equals zero there.
The height is opposite the 30° angle and 40 is the adjacent side, so tan 30° = h/40. Since tan 30° = 1/√3, h = 40/√3 = (40√3)/3.
The slope is (11 − 3)/(5 − 1) = 8/4 = 2. Using point (1, 3), point-slope form is y − 3 = 2(x − 1).
Order of operations: first 2² = 4, then 4 × 4 = 16 and 6 ÷ 3 = 2. So 3 + 16 − 2 = 17.
log₃(81) = 4 because 3⁴ = 81, and log₃(1/9) = −2 because 3⁻² = 1/9. Adding gives 4 + (−2) = 2.
Dividing p(x) by (x − 2) gives x² − 2x − 3, which factors as (x − 3)(x + 1). So the roots are 2, 3, and −1.
For |r| < 1 the infinite sum is a/(1 − r). Here a = 5 and r = −1/2, so S = 5/(1 + 1/2) = 5/(3/2) = 10/3.
The second equation is exactly twice the left side of the first but not twice the constant, so the lines have equal slopes and different intercepts. Parallel distinct lines never intersect, so the system is inconsistent.
Factoring the numerator gives (x − 2)(x + 2), so q(x) = x + 2 for x ≠ 2. The factor cancels, leaving a hole at x = 2 where the limit exists but the function is undefined.
The absolute value of a number is its distance from zero on the number line, so |−7| = 7 and |3| = 3. Adding these gives 7 + 3 = 10.
Combine like terms: 5x − 2x = 3x and 3y + y = 4y. The simplified expression is 3x + 4y.