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Quizzes > High School Quizzes > Electives

Solving Quadratic Equations: Complete Square Quiz

Master completing the square with interactive worksheets

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Complete the Square Challenge algebra quiz for high school students.

Express x^2 + 6x in completed square form.
(x + 3)^2 + 9
x(x + 6)
(x + 3)^2 - 9
(x - 3)^2 - 9
Completing the square for x^2 + 6x involves adding and subtracting (6/2)^2, which is 9, to form a perfect square. This results in the expression (x + 3)^2 - 9, which helps in identifying the vertex and solving related equations.
What constant must be added to x^2 + 4x to complete the square?
8
4
2
6
Dividing the coefficient of x (4) by 2 gives 2, and squaring 2 yields 4. This constant, 4, must be added to complete the square and form a perfect square trinomial.
Rewrite the quadratic expression x^2 - 10x in completed square form.
(x - 5)^2 - 25
(x - 5)^2 + 25
(x + 5)^2 - 25
(x + 5)^2 + 25
Halving the coefficient of x (-10) gives -5, and squaring -5 yields 25. Writing x^2 - 10x as (x - 5)^2 - 25 shows the vertex form of the quadratic, which is useful for further analysis.
Find the completed square form of x^2 + 8x.
(x - 4)^2 - 16
(x + 8)^2 - 4
(x + 4)^2 + 16
(x + 4)^2 - 16
By halving the coefficient of x (8), we obtain 4, and squaring 4 gives 16. Thus, the expression can be rewritten as (x + 4)^2 - 16, a form that makes it easier to identify important features of the quadratic.
Complete the square for the expression x^2 + 2x.
(x + 1)^2 - 1
(x + 1)^2 + 1
(x - 1)^2 - 1
(x + 2)^2 - 4
Halving the coefficient of x (2) gives 1, and squaring 1 results in 1. This allows us to express the quadratic as (x + 1)^2 - 1, which is a useful form for solving and graphing.
Rewrite 2x^2 + 12x + 3 in completed square form.
2(x + 3)^2 - 15
2(x + 3)^2 + 15
2(x - 3)^2 - 15
(x + 3)^2 - 15
First, factor out the 2 from the terms involving x to get 2(x^2 + 6x). Completing the square inside the parentheses by adding and subtracting 9 and then adjusting the constant yields 2(x + 3)^2 - 15.
Solve the quadratic equation x^2 + 6x + 5 = 0 by completing the square.
x = 1 and x = 5
x = -2 and x = -3
x = 2 and x = -8
x = -1 and x = -5
By rewriting the quadratic in the form (x + 3)^2 - 4 = 0 and then setting (x + 3)^2 equal to 4, we obtain x = -3 ± 2. This leads to the solutions x = -1 and x = -5, using the completing the square method.
Find the vertex form of y = x^2 - 8x + 15.
(x - 4)^2 - 1
(x - 4)^2 + 1
(x + 4)^2 - 1
(x - 4)^2 - 15
Completing the square for the quadratic x^2 - 8x transforms it into (x - 4)^2 - 16. Adding the constant 15 adjusts it to (x - 4)^2 - 1, which directly reveals the vertex at (4, -1).
Express 3x^2 + 9x + 2 in completed square form.
3(x + 3/2)^2 - 19/4
(x + 3)^2 - 19/4
3(x + 3/2)^2 + 19/4
3(x - 3/2)^2 - 19/4
After factoring out the 3 from the quadratic and linear terms, complete the square on (x^2 + 3x) by converting it into (x + 3/2)^2 - 9/4. Adjusting for the constant results in 3(x + 3/2)^2 - 19/4.
Express the quadratic x^2 - 4x - 12 in completed square form.
(x + 2)^2 - 16
(x - 2)^2 - 16
(x - 2)^2 + 16
(x - 2)^2 - 8
Completing the square for x^2 - 4x involves rewriting it as (x - 2)^2 - 4; adjusting for the -12 gives the final form (x - 2)^2 - 16. This form is valuable for identifying the vertex and graphing the quadratic.
For the quadratic equation x^2 + 10x + c = 0 to have exactly one real solution, what must c be?
25
30
20
16
A quadratic has a single real solution when it forms a perfect square, meaning its discriminant is zero. Completing the square shows c must equal (10/2)^2, which is 25.
Complete the square for the function y = 4x^2 - 16x + 7.
(x - 2)^2 - 9
4(x - 2)^2 - 9
4(x + 2)^2 - 9
4(x - 2)^2 + 9
After factoring out 4 from the x terms, the expression inside becomes x^2 - 4x. Completing the square yields 4(x - 2)^2 - 16, and adding 7 gives 4(x - 2)^2 - 9.
Rewrite the quadratic equation x^2 + 2x - 8 = 0 into completed square form and determine the value of (x + 1)^2.
1
9
7
8
Completing the square for x^2 + 2x gives (x + 1)^2 - 1; rewriting the equation results in (x + 1)^2 - 9 = 0. This implies (x + 1)^2 = 9, which is the value sought.
Find the vertex of the parabola given by y = -x^2 + 4x - 3 after completing the square.
(-2, -1)
(2, -1)
(2, 1)
(-2, 1)
By completing the square, the equation becomes -(x - 2)^2 + 1, which clearly shows that the vertex is at (2, 1). This form efficiently reveals the maximum point of the parabola.
Determine the completed square form of 5x^2 + 20x + 15.
5(x + 2)^2 - 5
5(x - 2)^2 - 5
5(x + 2)^2 + 5
5(x + 2)^2 - 3
Factoring out 5 gives 5(x^2 + 4x + 3). Completing the square on the expression inside yields 5[(x + 2)^2 - 1], which simplifies to 5(x + 2)^2 - 5. This form aids in further analysis of the quadratic.
Derive the completed square form for the quadratic equation 3x^2 - 12x + 11 and solve for its roots.
x = 2 ± 1/3
x = 2 ± √3
x = 2 ± (√3)/2
x = 2 ± 1/√3
Factoring 3 from the quadratic and linear terms yields 3(x^2 - 4x) + 11. Completing the square produces 3(x - 2)^2 - 1, and setting this equal to zero gives (x - 2)^2 = 1/3, leading to x = 2 ± 1/√3.
For the quadratic function f(x)= -2x^2 + 8x - 5, complete the square and determine its maximum value.
-3
5
3
8
Completing the square converts the function to -2(x - 2)^2 + 3. Since the coefficient of the squared term is negative, the parabola opens downward and its maximum value is 3, which occurs at x = 2.
Convert the quadratic 4x^2 + 12x + 9 into a perfect square form and state its vertex.
(2x - 3)^2 with vertex (-3/2, 0)
(2x - 3)^2 with vertex (3/2, 0)
(2x + 3)^2 with vertex (3/2, 0)
(2x + 3)^2 with vertex (-3/2, 0)
The quadratic factors neatly into (2x + 3)^2. Setting 2x + 3 = 0 gives x = -3/2, so the vertex is (-3/2, 0), confirming the perfect square form.
If a quadratic expression is given by ax^2 + bx + c and its completed square form is a(x - h)^2 + k, what is the expression for h in terms of a and b?
h = 2a/b
h = -b/(2a)
h = -2a/b
h = b/(2a)
When completing the square, the value h is determined by halving the coefficient of x (after factoring out a) and changing its sign, which gives h = -b/(2a). This formula is essential for rewriting a quadratic in vertex form.
Complete the square for the quadratic inequality x^2 - 6x + 8 ≤ 0 and identify the interval of x that satisfies the inequality.
x = 3
2 ≤ x ≤ 4
x ≤ 2 or x ≥ 4
1 ≤ x ≤ 5
Completing the square for x^2 - 6x yields (x - 3)^2 - 1, so the inequality becomes (x - 3)^2 ≤ 1. This means the distance from x to 3 is at most 1, hence x must lie between 2 and 4, inclusive.
0
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Study Outcomes

  1. Apply the completing the square method to rewrite quadratic equations in vertex form.
  2. Identify and isolate quadratic and linear components within an algebraic expression.
  3. Solve quadratic equations using the completed square technique with accuracy.
  4. Analyze the transformation process to simplify and verify solutions on standardized tests.
  5. Demonstrate the advantages of completing the square for graphing quadratic functions.

Quadratic Equations Quiz: Complete Square Cheat Sheet

  1. Understand the concept of completing the square - Completing the square lets you transform a pesky quadratic into a neat perfect square trinomial, making the solution practically jump off the page. This technique rewrites ax²+bx+c as (x+d)²=e to simplify your solving process. It's also the cornerstone for deriving the famous quadratic formula, so you'll see how everything connects! OpenStax: Solve Quadratics by Completing the Square
  2. Learn the b/2 formula trick - For any quadratic in the form x² + bx, add (b/2)² to both sides and watch the equation snap into a perfect square. This handy shortcut keeps your work neat and your mind sharp. Once you've mastered it, you'll breeze through problems that once felt daunting! OpenStax: Elementary Algebra - Completing the Square
  3. Practice with step-by-step examples - Drill yourself by isolating the quadratic term, adding the right constant, and solving for x every time. Regular practice turns confusing steps into second nature, boosting both speed and confidence. Make every problem a mini-victory on your path to mastering algebra! Neurochispas: Completing the Square Practice
  4. Recognize perfect square trinomials - Identifying patterns like x² + 6x + 9 = (x + 3)² saves you time and avoids errors. Once those "aha!" moments hit, you'll spot perfect squares in the blink of an eye. This skill is a true algebraic superpower that cuts down your work! Math Warehouse: Perfect Square Trinomials
  5. Derive the quadratic formula - By completing the square on ax²+bx+c=0, you unlock the universal quadratic formula and see why it works. This derivation deepens your understanding and ensures you're not just memorizing, but really learning. It's the ultimate proof that ties all your algebra skills together! OpenStax: Deriving the Quadratic Formula
  6. Explore the geometric interpretation - Visualize completing the square as rearranging area pieces to form a perfect square - geometry meets algebra in a fun mash-up! This perspective makes abstract steps feel concrete and intuitive. You'll never look at quadratics the same way again. OpenStax: Geometric View of Completing the Square
  7. Tackle non‑unit leading coefficients - When a≠1, factor it out first before adding (b/2)² and keep your algebra shipshape. This extra step ensures you're always on the right path, no matter how wild the coefficients get. With this trick in your toolkit, no quadratic can intimidate you! Neurochispas: Practice with a≠1
  8. Apply to real‑world problems - Use completing the square to maximize areas, optimize projectile paths, or even fine-tune financial models. Seeing math in action makes learning stick and shows you why these skills matter. Plus, solving practical puzzles is a blast - like algebra meets adventure! OpenStax: Real‑World Applications
  9. Memorize the four key steps - Isolate the quadratic term, add (b/2)², factor the trinomial, and solve for x. Locking in these steps turns each problem into a smooth, predictable routine. Before you know it, you'll tackle any quadratic with eyes closed! OpenStax: Key Steps Summary
  10. Build muscle memory with regular practice - Consistency is everything - solve a few problems daily to make completing the square second nature. Each session reinforces your skills, sharpens your instincts, and boosts your confidence. Stick with it, and quadratics will feel like a fun brain teaser instead of a chore! Neurochispas: Daily Practice Problems
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