A number is 2 less than its square. Find all such numbers.
Correct Answer:
The numbers that are 2 less than their square are –1 and 2.
Explanation (
Consider a number represented by a variable, say xxx. The statement “a number is 2 less than its square” can be expressed as a mathematical equation. Translating the statement gives:x=x2−2x = x^2 – 2x=x2−2
To solve this equation, rearrange the terms so that the equation equals zero:x2−x−2=0x^2 – x – 2 = 0x2−x−2=0
This is a quadratic equation in standard form. Solving such equations can be done using factoring. The goal is to find two numbers that multiply to give –2 and add to give –1.
The correct factorization is:(x−2)(x+1)=0(x – 2)(x + 1) = 0(x−2)(x+1)=0
Setting each factor equal to zero gives two possible solutions:x−2=0⇒x=2x – 2 = 0 \Rightarrow x = 2 x−2=0⇒x=2x+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1x+1=0⇒x=−1
Thus, the two numbers that satisfy the condition are –1 and 2.
Verification of each solution confirms correctness:
- For x=2x = 2x=2: the square is 22=42^2 = 422=4, and the number is 2, which is indeed 2 less than 4.
- For x=−1x = -1x=−1: the square is (−1)2=1(-1)^2 = 1(−1)2=1, and the number is –1, which is also 2 less than 1.
Both values meet the condition set by the problem statement. The process of solving involved recognizing a quadratic form, factoring accurately, and validating both solutions. The condition “a number is 2 less than its square” is satisfied only by these two values. Therefore, the complete solution set is {–1, 2}.
