Though the two cases look similar to the untrained eye, the perennial sin of treating the equation ax^{2}+bx+ac=0 and the function y=ax^{2}+bx+ac as the same thing has brought many students to grief (and blood-soaked test papers), especially after they anyhow bring terms left and right of the = sign when dealing with a function.
Obviously we're dealing with the second instant here as we need to find a, b and c in order to save Miss Loi's eyebrow, and we'll now attempt to dissect Ron's late night solution in a series of robust steps:
Step 1: Bring out coefficient of x^{2} and put the x and x^{2} terms inside brackets:
y = −(x^{2} − 3x) + 5
The coefficient of x^{2} is −1, so bringing it out will change the + 3x to − 3x inside the bracket (DUH! - yes simple algebra but then ...)
Step 2: + (half of x-coefficient)^{2} − (x-coefficient)^{2} within the brackets
y = −(x^{2} − 3x + (3/2)^{2} − (3/2)^{2}) + 5
Step 3: Move the new minus(−) term carefully out of the brackets, and be mindful of the need to multiply it with an constant outside the brackets.
y = −(x^{2} − 3x + (3/2)^{2}) + (3/2)^{2} + 5
Note that the −(3/2)^{2} has become a + (3/2)^{2} because of the − sign outside the brackets.
Tread carefully here, for this is careless mistake territory.
Step 4: Transform everything inside the brackets into a square term. Whatever constant that's outside the brackets, STAYS outside the brackets.
y = −(x − (3/2))^{2} + (3/2)^{2} + 5
Note that its (x − (3/2))^{2} NOT (x + (3/2))^{2} because of the negative − 3x within the brackets.
You're deep, deep into careless mistakes territory now ...
Step 5: Sum up the constant terms outside the brackets to obtain the final expression y = ±(x−p)^{2}+q, where (p, q) are the coordinates of the:
y = −(x − (3/2))^{2} + 7.25
Miss Loi's eyebrow - fierce isn't it?
And so from our final expression above, the maximum (since there's a -ve sign outside the brackets) point of the graph is (3/2, 7.25)
⇒ c = 7.25
⇒ b = 3/2 = 1.5
To find the y-intercept a, simply equate x to 0 to get a=5.
N.B. Performing the 5 steps above is completely safe whether you're completing the square for a function or an equation, as there's no moving of terms left and right of the = sign.
]]>Is this enough to tide you over miss loi?
cuts y-axis at y = 5, arch at x = 1.5, max value of y is at y=7.25.