Which value must be added to the expression [tex]x^2-3 x[/tex] to make it a perfect-square trinomial?
Answer (1)
We want to complete the square for the expression x 2 − 3 x .
Find a such that − 2 a = − 3 , which gives a = 2 3 .
Calculate a 2 = ( 2 3 ) 2 = 4 9 .
The value to be added is 4 9 .
Explanation
Understanding the Problem We want to find a value to add to the expression x 2 − 3 x to make it a perfect-square trinomial. A perfect-square trinomial has the form ( x + a ) 2 = x 2 + 2 a x + a 2 or ( x − a ) 2 = x 2 − 2 a x + a 2 . In our case, we have x 2 − 3 x , so we want to find a such that − 2 a = − 3 .
Solving for a To find the value of a , we solve the equation − 2 a = − 3 . Dividing both sides by − 2 , we get a = − 2 − 3 = 2 3 .
Calculating a^2 Now, we need to add a 2 to the expression to complete the square. So we calculate a 2 = ( 2 3 ) 2 = 2 2 3 2 = 4 9 .
The Perfect-Square Trinomial The value to be added to the expression is 4 9 . The perfect-square trinomial will be x 2 − 3 x + 4 9 = ( x − 2 3 ) 2 .
Final Answer Therefore, the value that must be added to the expression x 2 − 3 x to make it a perfect-square trinomial is 4 9 .
Examples
Completing the square is a technique used in various real-life applications, such as optimizing the design of parabolic reflectors or determining the trajectory of projectiles. For example, if you're designing a parabolic mirror for a solar oven, completing the square can help you find the optimal focal point to maximize the concentration of sunlight and improve the oven's efficiency. Similarly, in physics, completing the square can simplify equations related to projectile motion, making it easier to calculate the range and maximum height of a thrown object.
