Find two consecutive even integers such that the square of the smaller is 10 more than the larger.
x and (x+2)
x^2 - 10 = x+2
x^2 - x - 12 = 0
(x -4)(x+3) = 0
try x = 4
then 4 and 6
16 - 10 = ? 6 yes that checks
Let's assume the smaller even integer is x.
Step 1: Find the larger even integer:
Since the two integers are consecutive, we can find the larger even integer by adding 2 to the smaller integer. Therefore, the larger even integer is x + 2.
Step 2: Formulate the equation:
The square of the smaller integer is equal to 10 more than the larger integer. This can be written as:
x^2 = (x + 2) + 10
Step 3: Simplify the equation:
To solve for x, we simplify the equation by expanding the square:
x^2 = x + 12
Step 4: Rearrange the equation:
To get the equation in a standard form, we move all the terms to one side:
x^2 - x - 12 = 0
Step 5: Factor the equation (if possible):
By factoring the equation, we find the factors of -12 that add up to -1:
(x - 4)(x + 3) = 0
Step 6: Solve for x:
Setting each factor equal to zero, we can find the possible values of x:
x - 4 = 0 --> x = 4
x + 3 = 0 --> x = -3
Since we are looking for even integers, we discard the value -3 and consider x = 4.
Step 7: Find the two consecutive even integers:
Using the value of x, we determine the two consecutive even integers:
Smaller integer: x = 4
Larger integer: x + 2 = 4 + 2 = 6
Therefore, the two consecutive even integers are 4 and 6.
To find two consecutive even integers, let's assign variables to the integers. Let's say the smaller even integer is x, so the next consecutive even integer would be x + 2.
Now let's translate the problem into an equation. The square of the smaller integer is 10 more than the larger:
x^2 = (x + 2) + 10
To solve this equation, we can simplify it:
x^2 = x + 12
Rearranging the equation to bring all terms to one side:
x^2 - x - 12 = 0
Now we have a quadratic equation, and we can solve it by factoring or using the quadratic formula. In this case, the equation can be factored as:
(x - 4)(x + 3) = 0
Setting each factor equal to zero gives us two possible solutions:
x - 4 = 0 or x + 3 = 0
Solving these equations gives us:
x = 4 or x = -3
Since we are looking for even integers, we can disregard the negative solution (-3), leaving us with the smaller even integer being x = 4. So the two consecutive even integers are 4 and 6.
Therefore, the two consecutive even integers such that the square of the smaller is 10 more than the larger are 4 and 6.
Find two consecutive even integers such that the square of the smaller is 10
more than the larger