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A positive number, when added to its square, gives 56. The number is:
12
9
8
7
7
Look for two consecutive integers whose product is 56. Since 7├Ч8=56 and the equation is x(x+1)=56, the positive number x must be 7.
Let the positive number be x. The problem states x + x┬▓ = 56.
x2+xтИТ56=0
Look for two consecutive integers whose product is 56. Since 7├Ч8=56 and the equation is x(x+1)=56, the positive number x must be 7.
Students often misinterpret 'added to its square' as (x+x)2=56 or forget the condition that the number must be positive, potentially choosing -8 instead of 7.
Set up the equation
Let the positive number be x. According to the statement, the sum of the number and its square is 56: x+x2=56.
x2+xтИТ56=0
Factor the quadratic equation
We need two numbers that multiply to -56 and add to +1. These are 8 and -7. Thus, the equation becomes (x+8)(xтИТ7)=0.
(x+8)(xтИТ7)=0
Solve for x
Setting each factor to zero gives x=тИТ8 or x=7. Since the problem specifies a 'positive number', we exclude -8.
x=7
D is correct because 7 + 7┬▓ = 7 + 49 = 56, satisfying the condition given in the problem.
This concept of finding roots of a quadratic equation is fundamental to Algebra and appears frequently in physics problems involving kinematic equations where time or distance must be positive.
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