Megan wants to find the minimum value of the function \large y=x^2-10x+3 . She begins to convert the equation into vertex form by completing the square. Megan stops at the step below.
\large y=\left(x+a\right)^2+3+b
What number should Megan use to replace the \large a in this equation?
To convert a quadratic function from standard form to vertex form by completing the square, we need to follow these steps:
1. Group the x terms and the constant term.
2. Factor out the coefficient of the x^2 term, if necessary.
3. Complete the square by adding and subtracting the square of half of the coefficient of the x term.
4. Write the expression in vertex form by simplifying the squared expression and combining any constant terms.
Using these steps, we can start with the function y = x^2 - 10x + 3 and complete the square as follows:
1. y = (x^2 - 10x) + 3
2. y = (x^2 - 10x + 25) - 22 (note that we added and subtracted (-10/2)^2 = 25, which is equal to 5^2)
3. y = (x - 5)^2 - 22
4. y = (x - 5)^2 + (-22)
Now we can see that the equation is in vertex form, with vertex at (5, -22). Therefore, the number that Megan should use to replace the a in the equation y = (x + a)^2 + 3 + b is a = -5.