What is the first step when rewriting y=−4x2+2x−7 in the form y=a(x−h)2+k?

3 Answers
Douglas K.
Jul 3, 2017

There is a process for completing the square but the values, a,h,andk are far too easy to obtain by other methods. Please see the explanation.

Explanation:

  1. a=−4 the value of "a" is always the leading coefficient of the x2 term.
  2. h=−b2a=−22(−4)=14
  3. k=y(h)=y(14)=−4(14)2+2(14)−7=−274

This is a lot easier than adding zero to the original equation in the form of −4h2+4h2:

y=−4x2+2x−4h2+4h2−7

Removing a factor of -4 from the first 3 terms:

y=−4(x2−12x+h2)+4h2−7

Match the middle term of the expansion (x−h)2=x2−2hx+h2 with the middle term in the parenthesis:

−2hx=−12x

Solve for h:

h=14

Therefore, we can compress the 3 terms into (x−14)2:

y=−4(x−14)2+4h2−7

Substitute for h:

y=−4(x−14)2+4(14)2−7

Combine like terms:

y=−4(x−14)2−274

Look at how much easier is it to remember 3 simple facts.

hereyougo · EZ as pi
Jul 3, 2017

You would factor out the −4 from the first term giving you
y=−4(x2−12x)−7

Explanation:

First complete the square.
y=−4x2+2x−7
get the x2 term to have a coefficient of 1.
You can do this by factoring out −4 from the first two terms.
y=−4(x2−12x)−7
Then complete the square
y=−4(x−14)2−7−(116×−4)

this simplifies down to
y=−4(x−14)2−6.75

EZ as pi
Jul 3, 2017

Factor out −4 from each term, to get:

y=−4[x2−12x+74]

Explanation:

y=ax2+bx+c

In order to complete the square, the coefficient of x2 must be 1, so the first step will be to make this happen.

y=−4x2+2x−7 ← factor out −4 from each term to get:

y=−4[x2−12x+74]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For the sake of completeness the full process is shown below.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

y=−4[x2−12x +74] ← add and subtract (b2)2

b=−12 ⇒(b2)2=(−12÷2)2=(−14)2=116

y=−4[x2−12x+116−116+74]

y=−4[(x2−12x+116)+(−116+74)]

y=−4[(x−14)2+2716] ← distribute the −4

y=−4(x−14)2−274

y=−4(x−14)2−634

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