The product of two consecutive integers is 420. An equation is written in standard form to solve for the smaller integer by factoring. What is the constant of the quadratic expression in this equation?
[tex]x^2+x+\square=0[/tex]
Answer (2)
Let x be the smaller integer, so the next integer is x + 1 .
The product of the integers is x ( x + 1 ) = 420 , which expands to x 2 + x = 420 .
Rewrite in standard form: x 2 + x − 420 = 0 .
The constant term in the quadratic expression is − 420 .
Explanation
Problem Analysis Let's analyze the problem. We are given that the product of two consecutive integers is 420. We need to find the constant term of the quadratic equation in standard form that can be used to solve for the smaller integer.
Setting up the Equation Let x be the smaller integer. Then the next consecutive integer is x + 1 . The product of these two integers is given as 420. So, we can write the equation as: x ( x + 1 ) = 420
Expanding the Product Expanding the equation, we get: x 2 + x = 420
Standard Form To write the equation in standard form, we subtract 420 from both sides: x 2 + x − 420 = 0
Identifying the Constant Term Now, we compare this equation with the given form x 2 + x + □ = 0 . We can see that the constant term is -420.
Final Answer Therefore, the constant of the quadratic expression in the equation is -420.
Examples
Understanding quadratic equations is crucial in various real-life scenarios. For instance, when planning a garden, you might want to enclose a rectangular area with a specific amount of fencing. If you know the area you want to cover (e.g., 420 square feet) and the relationship between the length and width (e.g., the length is one foot more than the width), you can set up a quadratic equation to find the exact dimensions. Similarly, in physics, projectile motion problems often involve quadratic equations to determine the trajectory and landing point of an object, making this math concept highly applicable and practical.
The constant term of the quadratic expression formed by the product of two consecutive integers that equals 420 is -420. This comes from rearranging the equation into standard form. It is identified from the equation x 2 + x − 420 = 0 .
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