Question Complete the square to re-write the quadratic function in vertex form: $$ y=x^{2}-7 x+2 $$ Answer Attemptiout of 2

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UpStudy AI · Accepted Answer

1. Start with the quadratic function:  
   $$
   y = x^2 - 7x + 2
   $$

2. Group the $$ x $$-terms together:  
   $$
   y = \left(x^2 - 7x\right) + 2
   $$

3. To complete the square, take the coefficient of $$ x $$, which is $$-7$$. Half of $$-7$$ is $$-\frac{7}{2}$$. Square this value to get:  
   $$
   \left(-\frac{7}{2}\right)^2 = \frac{49}{4}
   $$

4. Add and subtract $$\frac{49}{4}$$ inside the parentheses:  
   $$
   y = \left(x^2 - 7x + \frac{49}{4} - \frac{49}{4}\right) + 2
   $$

5. Group the perfect square trinomial and combine the constant terms:  
   $$
   y = \left(x^2 - 7x + \frac{49}{4}\right) - \frac{49}{4} + 2
   $$

6. The trinomial $$\left(x^2 - 7x + \frac{49}{4}\right)$$ factors as:  
   $$
   \left(x - \frac{7}{2}\right)^2
   $$  
   So the function becomes:  
   $$
   y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 2
   $$

7. Express the $$ 2 $$ as $$\frac{8}{4}$$ and simplify:  
   $$
   y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + \frac{8}{4} = \left(x - \frac{7}{2}\right)^2 - \frac{41}{4}
   $$

8. The vertex form of the quadratic function is:  
   $$
   y = \left(x - \frac{7}{2}\right)^2 - \frac{41}{4}
   $$
The quadratic function $$ y = x^2 - 7x + 2 $$ in vertex form is $$ y = \left(x - \frac{7}{2}\right)^2 - \frac{41}{4} $$.

Question Complete the square to re-write the quadratic function in vertex form: \[ y=x^{2}-7 x+2 \] Answer Attemptiout of 2

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