Alessandro wrote the quadratic equation $-6=x^2+4 x-1$ in standard form. What is the value of $c$ in his new equation?
A. $c=-6$
B. $c=-1$
C. $c=5$
D. $c=7
Answer (2)
Rewrite the given equation − 6 = x 2 + 4 x − 1 in the standard form.
Add 6 to both sides: x 2 + 4 x − 1 + 6 = 0 .
Simplify the equation to x 2 + 4 x + 5 = 0 .
Identify the value of c as the constant term: 5 .
Explanation
Understanding the Problem We are given the quadratic equation − 6 = x 2 + 4 x − 1 . Our goal is to rewrite this equation in the standard form a x 2 + b x + c = 0 and then identify the value of c .
Rewriting in Standard Form To rewrite the equation in standard form, we need to move all terms to one side of the equation, leaving zero on the other side. We can do this by adding 6 to both sides of the equation: x 2 + 4 x − 1 + 6 = 0
Simplifying the Equation Now, we simplify the equation by combining the constant terms: x 2 + 4 x + 5 = 0
Identifying the Value of c In the standard form of a quadratic equation, a x 2 + b x + c = 0 , the coefficient c is the constant term. In our equation, x 2 + 4 x + 5 = 0 , we can see that c = 5 .
Examples
Understanding quadratic equations in standard form is crucial in many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown in the air. By knowing the standard form, they can easily identify the coefficients and constants needed to solve for key parameters like maximum height and range. Similarly, economists use quadratic equations to model cost and revenue functions, helping businesses optimize their production and pricing strategies. The constant term 'c' often represents fixed costs or initial investments, providing valuable insights for decision-making.
The value of c in Alessandro's rewritten quadratic equation is 5. This was determined by rearranging the original equation into standard form and identifying the constant term. Thus, the correct answer is C. c = 5 .
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