Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To write the quadratic function [tex]\( f(x) = 7x^2 + 42x \)[/tex] in vertex form, we need to follow a process called completing the square. Here is a step-by-step solution:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
2. Complete the square inside the parentheses:
- We look at the quadratic expression inside the parentheses: [tex]\( x^2 + 6x \)[/tex].
- To complete the square, we need to add and subtract a constant. This constant is found by taking half the coefficient of [tex]\( x \)[/tex] (which is 6 in this case), squaring it:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
- Add and subtract this constant inside the parentheses:
[tex]\[ x^2 + 6x = (x^2 + 6x + 9 - 9) = (x + 3)^2 - 9 \][/tex]
3. Rewrite the quadratic function with this completed square:
[tex]\[ f(x) = 7[(x + 3)^2 - 9] \][/tex]
4. Distribute the [tex]\( 7 \)[/tex] back into the equation:
[tex]\[ f(x) = 7(x + 3)^2 - 7 \cdot 9 \][/tex]
Simplify the constant term:
[tex]\[ 7 \cdot 9 = 63 \][/tex]
So,
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Thus, the function [tex]\( f(x) \)[/tex] written in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
The correct answer is:
[tex]\[ \boxed{7(x+3)^2-63} \][/tex]
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[ f(x) = 7(x^2 + 6x) \][/tex]
2. Complete the square inside the parentheses:
- We look at the quadratic expression inside the parentheses: [tex]\( x^2 + 6x \)[/tex].
- To complete the square, we need to add and subtract a constant. This constant is found by taking half the coefficient of [tex]\( x \)[/tex] (which is 6 in this case), squaring it:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
- Add and subtract this constant inside the parentheses:
[tex]\[ x^2 + 6x = (x^2 + 6x + 9 - 9) = (x + 3)^2 - 9 \][/tex]
3. Rewrite the quadratic function with this completed square:
[tex]\[ f(x) = 7[(x + 3)^2 - 9] \][/tex]
4. Distribute the [tex]\( 7 \)[/tex] back into the equation:
[tex]\[ f(x) = 7(x + 3)^2 - 7 \cdot 9 \][/tex]
Simplify the constant term:
[tex]\[ 7 \cdot 9 = 63 \][/tex]
So,
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
Thus, the function [tex]\( f(x) \)[/tex] written in vertex form is:
[tex]\[ f(x) = 7(x + 3)^2 - 63 \][/tex]
The correct answer is:
[tex]\[ \boxed{7(x+3)^2-63} \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.