Find the solutions of the quadratic equation [tex]$-9 x^2+49=0$[/tex].
A) [tex]$x=9 / 7,-9 / 7$[/tex]
B) [tex]$x=7 / 9,-7 / 9$[/tex]
C) [tex]$x=3 / 7,-7 / 3$[/tex]
D) [tex]$x=7 / 3,-7 / 3$[/tex]
Answer (1)
Rewrite the equation as 9 x 2 = 49 .
Divide both sides by 9 to get x 2 = 9 49 .
Take the square root of both sides: x = ± 9 49 .
Simplify the square root: x = ± 3 7 . The solutions are x = 3 7 and x = − 3 7 , so the answer is 3 7 , − 3 7 .
Explanation
Understanding the Problem We are given the quadratic equation − 9 x 2 + 49 = 0 and asked to find its solutions. This means we need to find the values of x that satisfy the equation.
Isolating the x^2 term First, let's rewrite the equation to isolate the x 2 term. We can add 9 x 2 to both sides of the equation to get: 49 = 9 x 2
Dividing by 9 Next, we divide both sides by 9 to get: x 2 = 9 49
Taking the Square Root Now, we take the square root of both sides of the equation. Remember to consider both positive and negative square roots: x = ± 9 49
Simplifying the Square Root Finally, we simplify the square root: x = ± 9 49 = ± 3 7
Final Answer So the solutions are x = 3 7 and x = − 3 7 . Therefore, the correct answer is D.
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the height of an object thrown into the air as a function of time. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic equation, and solving the equation can tell you when the ball will hit the ground.
