Complete the square to write $y=3 x^2+12 x+7$ in vertex form, $y=a(x-h)^2+k$.
$y=3\left(x^2+4 x\right)+1$
$y=3\left(x^2+4 x+4\right)+7$
When the above expression is written in vertex form, what are the values of a, h, and k?
Answer (2)
To rewrite the quadratic equation y = 3 x 2 + 12 x + 7 in vertex form:
Factor out 3: y = 3 ( x 2 + 4 x ) + 7 .
Complete the square: y = 3 ( x 2 + 4 x + 4 ) + 7 − 12 .
Rewrite in vertex form: y = 3 ( x + 2 ) 2 − 5 .
Identify the parameters: a = 3 , h = − 2 , k = − 5 .
Explanation
Understanding the Problem We are given the quadratic equation y = 3 x 2 + 12 x + 7 and we want to rewrite it in vertex form, which is y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola.
Factoring First, factor out the coefficient of the x 2 term (which is 3) from the x 2 and x terms: y = 3 ( x 2 + 4 x ) + 7
Completing the Square To complete the square inside the parenthesis, we need to add and subtract a value that will make the expression inside the parenthesis a perfect square. We take half of the coefficient of the x term (which is 4), square it (which is 2 2 = 4 ), and add it inside the parenthesis. To keep the equation balanced, we must subtract the same value outside the parenthesis. However, since we added 4 inside the parenthesis which is being multiplied by 3, we are actually adding 3 × 4 = 12 to the expression. Therefore, we must subtract 12 outside the parenthesis: y = 3 ( x 2 + 4 x + 4 ) + 7 − 12
Rewriting as a Squared Term Now, rewrite the expression inside the parenthesis as a squared term: y = 3 ( x + 2 ) 2 − 5
Identifying a, h, and k Comparing the equation y = 3 ( x + 2 ) 2 − 5 with the vertex form y = a ( x − h ) 2 + k , we can identify the values of a , h , and k :
a = 3
Since we have ( x + 2 ) in our equation, we can rewrite it as ( x − ( − 2 )) . Therefore, h = − 2 .
k = − 5
Final Answer Therefore, when the expression is written in vertex form, a = 3 , h = − 2 , and k = − 5 .
Examples
Completing the square is a useful technique in physics, especially when dealing with projectile motion. For example, if you have an equation that describes the height of a ball thrown in the air as a function of time, you can complete the square to find the maximum height of the ball and the time at which it reaches that height. This allows you to analyze and predict the behavior of the projectile, such as determining its range and time of flight. Understanding how to rewrite quadratic equations in vertex form helps in optimizing various real-world scenarios, from maximizing the area of a garden to designing efficient parabolic reflectors.
By completing the square for the equation y = 3 x 2 + 12 x + 7 , we rewrite it in vertex form as y = 3 ( x + 2 ) 2 − 5 . The values identified are a = 3 , h = − 2 , and k = − 5 . This method helps in understanding the vertex of the parabola represented by the quadratic equation.
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