\( x^{2}=6 x+1 \) 1) Rewrite the equation by completing the square. Your equation should look like \( (x+a)^{2}=b \) or \( (x-c)^{2}=d \). .

To complete the square for the equation \( x^{2} = 6x + 1 \), first, rearrange it: \[ x^{2} - 6x = 1. \] Now, take half of the coefficient of \( x \) (which is -6), square it, and add it to both sides: \[ \left( \frac{-6}{2} \right)^{2} = 9. \] Now, rewrite the equation: \[ x^{2} - 6x + 9 = 1 + 9, \] \[ (x - 3)^{2} = 10. \] Thus, the completed square form of the equation is \( (x - 3)^{2} = 10 \). Once you've completed the square, it's essential to understand how this method helps in visualizing the quadratic graph. The equation represents a parabola with its vertex at the point \( (3, 10) \), providing insight into its maximum/minimum value. As you tackle completing the square, watch out for those pesky arithmetic errors while squaring and adding! It's easy to mix up signs or forget to add the same number to both sides. Quick tips: double-check your half-coefficient calculation, and remember to add/subtract correctly—your future self will thank you!