Represent algebraically: A number squared, increased by 15, is the same as the square of one more than the number. My teacher said x^2 + 15 = x^2 + 1 Is incorrect.
The Correct Answer and Explanation is:
To represent the statement “A number squared, increased by 15, is the same as the square of one more than the number” algebraically, let the number be represented by xxx.
- First part of the statement: “A number squared, increased by 15” means the square of xxx plus 15, which is written as: x2+15x^2 + 15×2+15
- Second part of the statement: “The square of one more than the number” means the square of x+1x + 1x+1. This is written as: (x+1)2(x + 1)^2(x+1)2
Now, according to the problem, these two expressions are equal. So, we set them equal to each other:x2+15=(x+1)2x^2 + 15 = (x + 1)^2×2+15=(x+1)2
Expanding the right-hand side:
We need to expand (x+1)2(x + 1)^2(x+1)2. This is done by applying the binomial expansion:(x+1)2=x2+2x+1(x + 1)^2 = x^2 + 2x + 1(x+1)2=x2+2x+1
Setting up the equation:
Substitute the expanded form of (x+1)2(x + 1)^2(x+1)2 back into the equation:x2+15=x2+2x+1x^2 + 15 = x^2 + 2x + 1×2+15=x2+2x+1
Simplifying the equation:
Now, subtract x2x^2×2 from both sides to eliminate the x2x^2×2 terms:15=2x+115 = 2x + 115=2x+1
Solving for xxx:
Next, subtract 1 from both sides:14=2×14 = 2×14=2x
Finally, divide by 2 to solve for xxx:x=7x = 7x=7
Conclusion:
The correct algebraic representation of the problem is:x2+15=(x+1)2x^2 + 15 = (x + 1)^2×2+15=(x+1)2
The incorrect equation your teacher pointed out, x2+15=x2+1x^2 + 15 = x^2 + 1×2+15=x2+1, fails to account for the relationship between the number and one more than it, as it ignores the linear and constant terms that come from expanding (x+1)2(x + 1)^2(x+1)2. Therefore, the correct algebraic equation involves expanding (x+1)2(x + 1)^2(x+1)2 as shown.
