The axis of symmetry for a quadratic equation can be found using the formula [tex]$x=\frac{-b}{2 a}$[/tex], where [tex]$a$[/tex] and [tex]$b$[/tex] are coefficients in the quadratic equation and [tex]$x$[/tex] represents the values along a vertical line on the coordinate plane.
What is the equation when solved for [tex]$a$[/tex]?
A. [tex]$a=\frac{b}{2 x}$[/tex]
B. [tex]$a=\frac{-b}{2 x}$[/tex]
C. [tex]$a=\frac{1}{2} b x$[/tex]
D. [tex]$a=-\frac{1}{2} b x$[/tex]
Answer (2)
The equation for the axis of symmetry x = 2 a − b can be rearranged to solve for a , yielding a = 2 x − b . Therefore, the correct answer from the options given is B. This skill is crucial for solving equations in various mathematical contexts.
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Start with the equation x = 2 a − b .
Multiply both sides by 2 a to get 2 a x = − b .
Divide both sides by 2 x to isolate a .
The equation solved for a is a = 2 x − b .
Explanation
Understanding the Problem We are given the equation for the axis of symmetry of a quadratic equation:
Stating the Equation x = 2 a − b
Isolating 'a' Our goal is to solve this equation for a . To do this, we will first multiply both sides of the equation by 2 a :
Multiplying by 2a 2 a x = − b
Dividing by 2x Next, we will divide both sides of the equation by 2 x to isolate a :
Solving for a a = 2 x − b
Final Answer Therefore, the equation solved for a is:
Stating the Solution a = 2 x − b
Examples
Understanding how to rearrange equations like this is useful in many areas of math and science. For example, in physics, you might need to rearrange the formula for force, F = ma , to solve for acceleration, a = F / m . Similarly, in economics, you might rearrange the supply and demand equations to find the equilibrium price and quantity. These skills are essential for problem-solving in various fields.
