What are the solutions of $\frac{1}{4} x^2=-\frac{1}{2} x+2$?
A. -4 and 2
B. -4 and 1
C. 0 and 4
D. 1 and 4
Answer (1)
Rewrite the equation in standard quadratic form: x 2 + 2 x − 8 = 0 .
Factor the quadratic expression: ( x + 4 ) ( x − 2 ) = 0 .
Set each factor to zero and solve for x : x + 4 = 0 or x − 2 = 0 .
The solutions are x = − 4 and x = 2 , so the final answer is − 4 and 2 .
Explanation
Problem Analysis We are given the quadratic equation 4 1 x 2 = − 2 1 x + 2 . Our goal is to find the solutions for x .
Eliminating Fractions First, let's rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . To do this, we can start by multiplying both sides of the equation by 4 to eliminate the fractions: 4 × 4 1 x 2 = 4 × ( − 2 1 x + 2 ) x 2 = − 2 x + 8
Standard Quadratic Form Next, we move all the terms to the left side of the equation to set it equal to zero: x 2 + 2 x − 8 = 0
Factoring Now, we need to factor the quadratic expression x 2 + 2 x − 8 . We are looking for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2.
Factored Equation We can rewrite the quadratic equation as: ( x + 4 ) ( x − 2 ) = 0
Solving for x To find the solutions for x , we set each factor equal to zero and solve for x :
x + 4 = 0 or x − 2 = 0
Solutions Solving for x in each case gives us: x = − 4 or x = 2
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. They are also used in engineering to design bridges and arches, ensuring structural stability. In finance, quadratic equations can help calculate investment returns and model growth rates, providing valuable insights for financial planning.
