Rewrite quadratics, solve equations, and identify vertices using the completing-the-square method.
20 varied completing-the-square questions with instant worked feedback.
After the test · quadratic transformation lab
This practice set treats completing the square as a general quadratic tool. The method is used to create perfect-square trinomials, convert standard form to vertex form, solve equations, identify vertices and extreme values, work with leading coefficients and fractions, and determine parameters that produce perfect squares. The essential safeguard is compensation: any quantity introduced to create the square must be balanced so the new expression remains equivalent to the original.
Four-step method
The half-and-square step must use the x-coefficient inside the bracket.
This creates the missing constant for a perfect-square trinomial.
If a leading coefficient sits outside the bracket, the compensation must account for it.
This catches sign and compensation errors before choosing an answer.
Quick formula shelf
x² + bx + (b/2)² = (x + b/2)²
Halve the coefficient of x, then square that half.
y = a(x − h)² + k
The vertex is (h, k).
ax² + bx = a[x² + (b/a)x]
The coefficient to halve is b/a inside the bracket.
(x − h)² = r → x = h ± √r
Keep both square-root branches unless r = 0.
y = a(x − h)² + k
If a > 0, k is the minimum value; if a < 0, k is the maximum value.
x² + bx + c is a perfect square when c = (b/2)²
This directly supports parameter questions described on the page.
Visual compensation balance
The +9 is chosen because half of 6 is 3, and 3² = 9.
Adding and subtracting 9 keeps the rewritten expression equivalent to x² + 6x.
Half-and-square engine
The x-coefficient is 8.
Do not use the full coefficient.
This is the missing constant.
The binomial sign follows the sign of the half-coefficient.
Visual vertex form
y = a(x − h)² + k
(h, k). Be careful: x + 3 means h = −3 because x + 3 = x − (−3).
If a > 0, the parabola opens upward and k is a minimum. If a < 0, it opens downward and k is a maximum.
Solving by completing the square
Put the constant on the opposite side.
Add 9 to both sides.
The left side is now a perfect square.
Continue to both solution branches.
Fractional coefficients
Half of 3/2 is 3/4.
Keep the rational arithmetic exact.
Parameters and missing constants
c = (10/2)² = 25 creates a perfect-square trinomial.
Expanding verifies the parameter choice.
The completion constant is still positive; the binomial sign follows the linear term.
Vertices and extreme values
This matches the page's stated use of completing the square for vertices, extrema, and short applications.
y = 2(x − 3)² − 5
Because 2 > 0, the parabola opens upward. The vertex is (3, −5), so the minimum value is −5.
y = −3(x + 2)² + 7
Because −3 < 0, the parabola opens downward. The vertex is (−2, 7), so the maximum value is 7.
Common mistakes from the page
The method always halves the relevant x-coefficient first.
The two operations are separate: halve, then square.
Expanding the proposed square is a fast sign check.
The leading coefficient multiplies the inside compensation.
When solving after completing the square, both square-root branches must be retained unless the right side is zero.
Final completing-the-square checklist