1. ### The product of two consecutive integers is greater than 100

The product of two consecutive integers is greater than 100 Take an integer x. Next consecutive integer is x + 1 The product of those integers is: x(x + 1) This product is greater than 100 which gives us the algebraic expression of: x(x + 1) > 100 IF we want to solve for x: x^2 + x > 100...
2. ### Quadratic equation hacks using the discriminant

Quadratic equation hacks using the discriminant Solve x^2- 4x+ 5 using a discriminant: Discriminant is: Discriminant = b^2- 4ac Discriminant = (-4)^2 - 4(1)(5) Discriminant = 16 - 20 Discriminant = -4 When Discriminant < 0, the quadratic has no solution
3. ### your classmate asserted that x^2 - 4x - 12 and 12 - 4x - x^2 has the same factors is your classmate

your classmate asserted that x^2 - 4x - 12 and 12 - 4x - x^2 has the same factors is your classmate correct Factor x^2 - 4x - 12 using binomials: (x + 2)(x - 6) Therefore, factors are x = -2, x = 6 Factor 12 - 4x - x^2 -(x - 6)(x + 2) Therefore, factors are x = -2, x = -6 Because they don't...
4. ### difference between 2 positive numbers is 3 and the sum of their squares is 117

difference between 2 positive numbers is 3 and the sum of their squares is 117 Declare variables for each of the two numbers: Let the first variable be x Let the second variable be y We're given 2 equations: x - y = 3 x^2 + y^2 = 117 Rewrite equation (1) in terms of x by adding y to each...
5. ### x squared plus a minus b

x squared plus a minus b x squared means we raise x to the power of 2: x^2 Plus a: x^2 + a Minus b: x^2 + a - b
6. ### Divide 73 into two parts whose product is 402

Divide 73 into two parts whose product is 40 Our first part is x Our second part is 73 - x The product of the two parts is: x(73 - x) = 40 Multiplying through, we get: -x^2 + 73x = 402 Subtract 40 from each side, we get: -x^2 + 73x - 402 = 0 This is a quadratic equation. To solve this, we...
7. ### If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find th

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find the time taken by aeroplane to cover 1200m initially. We know from the distance formula (d) using rate (r) and time (t) that: d = rt Regular speed: 1200 = rt Divide each side by t, we get: r =...
8. ### A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an ho

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? Set up the relationship of still water speed and downstream speed Speed down stream = Speed in still water + speed of the...
9. ### if a number is added to its square, it equals 20

if a number is added to its square, it equals 20. Let the number be an arbitrary variable, let's call it n. The square of the number means we raise n to the power of 2: n^2 We add n^2 to n: n^2 + n It equals 20 so we set n^2 + n equal to 20 n^2 + n = 20 This is a quadratic equation. So we...
10. ### Sum of a number and it's reciprocal is 6. What is the number?

Sum of a number and it's reciprocal is 6. What is the number? Let the number be n. The reciprocal is 1/n. The word is means an equation, so we set n + 1/n equal to 6 n + 1/n = 6 Multiply each side by n to remove the fraction: n^2 + 1 = 6n Subtract 6n from each side: n^2 - 6n + 1 = 0 <-- This...
11. ### One number is equal to the square of another. Find the numbers if both are positive and their sum is

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650 Let the number be n. Then the square is n^2. We're given: n^2 + n = 650 Subtract 650 from each side: n^2 + n - 650 = 0 We have a quadratic equation. We type this into our search engine and...
12. ### if a number is added to its square, the result is 72. find the number

if a number is added to its square, the result is 72. find the number. Let the number be n. We're given: n + n^2 = 72 Subtract 72 from each side, we get: n^2 + n - 72 = 0 This is a quadratic equation. We type this equation into our search engine, and we get: n = 8 and n = -9
13. ### a rectangle has an area of 238 cm 2 and a perimeter of 62 cm. What are its dimensions?

a rectangle has an area of 238 cm 2 and a perimeter of 62 cm. What are its dimensions? We know the rectangle has the following formulas: Area = lw Perimeter = 2l + 2w Given an area of 238 and a perimeter of 62, we have: lw = 238 2(l + w) = 62 Divide each side of (1) by w: l = 238/w...
14. ### When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negati

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution. Let the number be n. Square of a number: n^2 28 is subtracted from the square of a number: n^2 - 28 3 times the number: 3n The result is mean an equation, so we set n^2 - 28 =...
15. ### Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which

Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which equation could be used to find Jeremy's age, j, if he is the younger man. Let Sam's age be s. Let' Jeremy's age be j. We're given: s = j + 2 <-- consecutive odd integers sj = 783 Substitute...
16. ### If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integ

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integer. Let the integer be "x". Square the integer: x^2 7 times the square: 7x^2 5 times the integer: 5x Add them together: 7x^2 + 5x The result is means an equation, so we set 7x^2 + 5x equal to 2...
17. ### When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positiv

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution. Let the number be n. We have: n^2 - 4 = 3n Subtract 3n from each side: n^2 - 3n - 4 = 0 Typing this quadratic equation into the search engine, we get: n = (-1, 4) The problem asks...
18. ### The Square of a positive integer is equal to the sum of the integer and 12. Find the integer

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer Let the integer be x. The sum of the integer and 12 is written as x + 12. The square of a positive integer is written as x^2. We set these equal to each other: x^2 = x + 12 Subtract x + 12 from each...
19. ### Pleasantburg has a population growth model of P(t)=at2+bt+P0 where P0 is the initial population. Sup

Pleasantburg has a population growth model of P(t)=at^2+bt+P0 where P0 is the initial population. Suppose that the future population of Pleasantburg t years after January 1, 2012, is described by the quadratic model P(t)=0.7t^2+6t+15,000. In what month and year will the population reach 19,200...
20. ### When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 intege

When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 integers. What is the difference of the product minus the sum? Let the 3 consecutive positive integers be: x x + 1 x + 2 The product is: x(x + 1)(x + 2) The sum is: x + x + 1 + x + 2 = 3x + 3 We're...