IDNLearn.com: Your reliable source for finding expert answers. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.
Sagot :
To rewrite the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's follow these steps carefully:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
2. Complete the square inside the parenthesis:
- Take the coefficient of [tex]\( x \)[/tex], which is 6, and halve it to get 3.
- Square 3 to get [tex]\( 3^2 = 9 \)[/tex].
So, we can rewrite the trinomial by adding and subtracting [tex]\( 9 \)[/tex] inside the parenthesis:
[tex]\[ f(x) = 7(x^2 + 6x + 9 - 9) \][/tex]
[tex]\[ f(x) = 7((x^2 + 6x + 9) - 9) \][/tex]
[tex]\[ f(x) = 7((x + 3)^2 - 9) \][/tex]
3. Distribute the 7 across the terms inside the parenthesis:
[tex]\[ f(x) = 7(x + 3)^2 - 7 \cdot 9 \][/tex]
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Therefore, the function [tex]\( f(x) = 7x^2 + 42x \)[/tex] written in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Hence, the correct answer is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{f(x)=7(x+3)^2-63} \][/tex]
Let's follow these steps carefully:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
2. Complete the square inside the parenthesis:
- Take the coefficient of [tex]\( x \)[/tex], which is 6, and halve it to get 3.
- Square 3 to get [tex]\( 3^2 = 9 \)[/tex].
So, we can rewrite the trinomial by adding and subtracting [tex]\( 9 \)[/tex] inside the parenthesis:
[tex]\[ f(x) = 7(x^2 + 6x + 9 - 9) \][/tex]
[tex]\[ f(x) = 7((x^2 + 6x + 9) - 9) \][/tex]
[tex]\[ f(x) = 7((x + 3)^2 - 9) \][/tex]
3. Distribute the 7 across the terms inside the parenthesis:
[tex]\[ f(x) = 7(x + 3)^2 - 7 \cdot 9 \][/tex]
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Therefore, the function [tex]\( f(x) = 7x^2 + 42x \)[/tex] written in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Hence, the correct answer is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{f(x)=7(x+3)^2-63} \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.