Which of the following equations has both 2 and -4 as solutions?
a. $x^2+6 x+8=0$
b. $x^2-2 x-8=0$
c. $x^2+2 x-8=0$
d. $x^2-2 x+8=0$
Answer (1)
Form the quadratic equation with roots 2 and -4 as ( x − 2 ) ( x + 4 ) = 0 .
Expand the equation: x 2 + 4 x − 2 x − 8 = 0 , which simplifies to x 2 + 2 x − 8 = 0 .
Compare the resulting equation with the given options.
The equation with both 2 and -4 as solutions is x 2 + 2 x − 8 = 0 .
Explanation
Understanding the Problem We are given four quadratic equations and asked to find the one that has both 2 and -4 as solutions. This means that when we substitute x = 2 and x = -4 into the correct equation, the equation will be true (equal to 0).
Setting up the Equation A quadratic equation with roots 2 and -4 can be written in the form ( x − 2 ) ( x + 4 ) = 0 . This is because if x = 2 , then ( 2 − 2 ) ( 2 + 4 ) = 0 , and if x = − 4 , then ( − 4 − 2 ) ( − 4 + 4 ) = 0 .
Expanding the Equation Now, let's expand the expression ( x − 2 ) ( x + 4 ) :
( x − 2 ) ( x + 4 ) = x 2 + 4 x − 2 x − 8 = x 2 + 2 x − 8
So, the quadratic equation is x 2 + 2 x − 8 = 0 .
Comparing with Options Now, we compare the resulting equation x 2 + 2 x − 8 = 0 with the given options: a. x 2 + 6 x + 8 = 0 b. x 2 − 2 x − 8 = 0 c. x 2 + 2 x − 8 = 0 d. x 2 − 2 x + 8 = 0
We can see that option c, x 2 + 2 x − 8 = 0 , matches our equation.
Final Answer Therefore, the equation that has both 2 and -4 as solutions is x 2 + 2 x − 8 = 0 .
Examples
In engineering, finding the solutions to a quadratic equation is crucial for designing stable structures. For example, when calculating the load a bridge can withstand, engineers solve quadratic equations to determine critical points where the structure might be at risk of failure. The roots of the equation (like 2 and -4 in our problem) represent these critical values, helping engineers ensure the bridge's stability and safety under various loads. This ensures public safety by preventing collapses and optimizing structural design.
