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Quizzes > High School Quizzes > Electives

Quadratic Equations Practice Quiz: Completing the Square

Boost confidence with completing the square practice problems.

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Square Completion Challenge, a high school algebra quiz.

When completing the square for the expression x² + 8x, what number should be added and subtracted to form a perfect square?
64
4
16
8
Half of 8 is 4 and squaring 4 gives 16. This value is added and subtracted to complete the square.
Rewrite x² + 6x + 5 in vertex form by completing the square. What is the correct form?
(x - 3)² - 4
(x + 3)² - 5
(x + 3)² - 4
(x + 3)² + 4
Completing the square for x² + 6x gives (x + 3)² after adding 9, and adjusting with the original constant results in (x + 3)² - 4.
Why is it important to factor out the coefficient of x² when it is not 1 during the process of completing the square?
It is not necessary
To simplify the constant term
To eliminate the x term
To make the coefficient of x² equal to 1
Factoring out the coefficient ensures that the quadratic term has a coefficient of 1, which simplifies the arithmetic involved in forming a perfect square.
What is the vertex of the parabola described by the equation y = (x + 4)² - 7?
(-7, 4)
(4, -7)
(-4, -7)
(7, -4)
The vertex form y = (x + 4)² - 7 indicates that the x-coordinate is -4 and the y-coordinate is -7. Recognizing the form (x - h)² + k helps identify the vertex.
In completing the square for the quadratic equation x² - 4x + 1 = 0, what value must be added inside the grouping to create a perfect square?
2
-2
4
-4
Half of -4 is -2, and squaring -2 gives 4. This number is necessary to complete the square and form the binomial square.
Complete the square for 2x² + 8x + 5 = 0. What is the equation in completed square form?
2(x + 2)² = 3
2(x + 2)² = -3
(x + 2)² = 3
2(x - 2)² = 3
Factoring 2 from the quadratic and linear terms yields 2(x² + 4x). Completing the square inside gives 2[(x + 2)² - 4] + 5, which simplifies to 2(x + 2)² - 3 = 0, or 2(x + 2)² = 3.
Solve x² + 10x + 21 = 0 by completing the square.
x = 3 or x = 7
x = -1 or x = -21
x = 2 or x = -2
x = -3 or x = -7
Completing the square transforms the equation into (x + 5)² = 4. Taking square roots gives x + 5 = ±2, resulting in the solutions x = -3 and x = -7.
After adding (b/2)² to both sides for an equation of the form x² + bx = c, what is the next step in completing the square?
Factor the left-hand side as a perfect square
Multiply both sides by 2
Add the constant term c to both sides
Subtract (b/2)² from both sides
Once (b/2)² is added to both sides, the left-hand side becomes a perfect square trinomial. Factoring it produces a squared binomial, which is essential before solving for x.
How does expressing a quadratic function in completed square (vertex) form help in analyzing its graph?
It eliminates the need to calculate the derivative
It directly reveals the vertex through the form (x + d)² + k
It makes finding the y-intercept easier
It provides the quadratic's factorization
Rewriting the quadratic in vertex form, (x + d)² + k, shows the vertex of the parabola immediately. This information is valuable for graphing and understanding the function's maximum or minimum.
Convert y = -x² + 6x - 8 into vertex form by completing the square.
y = -(x - 3)² + 1
y = (x - 3)² + 1
y = -(x - 3)² - 1
y = -(x + 3)² + 1
Factoring out the negative sign from the quadratic part and completing the square yields y = -(x - 3)² + 1. This form reveals the vertex of the parabola as (3, 1).
After completing the square for a quadratic equation, what is the standard method to solve for x?
Multiply both sides by x
Take the square root of both sides and solve the resulting equations
Expand the squared term and equate coefficients
Set the squared term equal to zero
Once the equation is written in the form (x + d)² = e, the next step is to take the square root of both sides. This procedure produces two linear equations, which are then solved for x.
When solving 3x² + 12x + 7 = 0 by completing the square, what is the essential initial step?
Ignore the coefficient of x²
Factor out 3 from the quadratic and linear terms
Divide the constant term by 3 only
Multiply x² by 3 to complete the square
Factoring 3 from the quadratic and linear terms ensures that the x² coefficient becomes 1, simplifying the process of completing the square.
Find the minimum value of the function f(x) = x² - 8x + 15 by completing the square.
15
1
-1
16
Completing the square transforms the function to (x - 4)² - 1. Since the squared term is always nonnegative, the minimum value occurs when (x - 4)² is 0, resulting in a minimum value of -1.
When completing the square for a quadratic with fractional coefficients, what additional step is often necessary?
Add the fraction as is to both sides
Factor out the fractional coefficient from the quadratic and linear terms
Ignore the fractional part during completion
Multiply only the constant term by the fraction
Factoring out the fractional coefficient ensures that the x² term has a coefficient of 1, which is key to correctly completing the square. This step simplifies the calculation.
Solve 4x² - 12x + 9 = 0 by completing the square.
x = 3/2
x = -3
x = -3/2
x = 3
Recognizing that 4x² - 12x + 9 is a perfect square trinomial, it can be written as (2x - 3)² = 0. Solving this yields the unique solution x = 3/2.
Solve by completing the square: 5x² + 20x - 15 = 0.
x = 2 ± √7
x = -2 ± √3
x = 2 ± √3
x = -2 ± √7
Dividing the equation by 5 simplifies it to x² + 4x - 3 = 0. Completing the square gives (x + 2)² = 7, from which taking the square root yields x = -2 ± √7.
Express 6x² - x - 2 = 0 in vertex form by completing the square.
6(x + 1/12)² - 49/24
6(x - 1/12)² + 49/24
6(x - 1/12)² - 49/24
6(x - 1/6)² - 49/24
First factor out 6 from the x terms to obtain 6(x² - (1/6)x) - 2 = 0. Completing the square inside the parentheses and adjusting the constant transforms the equation into the vertex form 6(x - 1/12)² - 49/24.
Solve -2x² + 4x + 6 = 0 by completing the square.
x = -1 or x = -3
x = 3 or x = -1
x = -3 or x = 1
x = 2 or x = -2
Multiplying the equation by -1 transforms it to 2x² - 4x - 6 = 0, and dividing by 2 gives x² - 2x - 3 = 0. Completing the square leads to (x - 1)² = 4, so x = 1 ± 2, which results in x = 3 or x = -1.
What is a necessary initial step when completing the square for an equation with a negative leading coefficient?
Square the negative coefficient
Factor out the negative coefficient from the quadratic and linear terms
Ignore the negative sign
Add a positive constant to both sides
When the leading coefficient is negative, factoring it out converts the quadratic into a form with a positive x² coefficient. This is essential for accurately completing the square.
Solve the quadratic equation x² + 5x + 6 = 0 by completing the square. What are the roots?
x = -2 and x = -3
x = 1 and x = 6
x = 2 and x = 3
x = -1 and x = -6
Completing the square for x² + 5x transforms the equation into (x + 2.5)² = 0.25. Taking the square root leads to x = -2.5 ± 0.5, which simplifies to the roots x = -3 and x = -2.
0
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Study Outcomes

  1. Apply the completing the square method to solve quadratic equations.
  2. Analyze quadratic expressions to rewrite them in vertex form.
  3. Determine critical values and identify the vertex of a parabola using algebraic techniques.
  4. Verify solutions by back-substituting into the original equation.

9-5 Quadratics Quiz: Complete the Square Cheat Sheet

  1. Recognize the standard quadratic form - Every quadratic equation starts as ax² + bx + c = 0, so spotting a, b and c helps you prep for completing the square without confusion. Getting comfortable with the letters means you can jump right into rearranging terms and adding the magic (b/2)². OpenStax guide
    2. OpenStax guide
  2. Break down the completing-the-square steps - Start by making sure the x²-coefficient is 1, move the constant to the other side, then add (b/2)² to craft a perfect square trinomial, factor it, and solve using square roots. Following these bite-size steps will make the method feel more like a fun puzzle than a chore. Stick to the order and you'll avoid mistakes. Step-by-step outline
    3. Step-by-step outline
  3. Handle non‑unit x² coefficients - If the coefficient of x² isn't 1, simply divide the entire equation by that number before you start adding squares. This trick clears away extra factors and keeps your perfect square neat and tidy. Example walkthrough
    4. Example walkthrough
  4. Maintain equality by adding to both sides - Completing the square means you must add the same (b/2)² value to both sides to keep the balance of the equation intact. Think of it like a scale: if you add weight on one side, you must match it on the other. This principle is the backbone of the method. Equality explained
    5. Equality explained
  5. Expect complex solutions - When your perfect square trinomial leads to a negative number under the square-root, you're diving into complex numbers. Don't panic - this just means your solutions will involve "i" and can be tackled the same way, with an extra splash of imaginary fun. Complex case guide
    6. Complex case guide
  6. Get hands-on practice - Mastery comes from doing, so jump into various quadratic puzzles to build your muscle memory. A variety of problems means you'll see patterns faster and tackle any twist the equation world throws at you. MathBitsNotebook practice problems
    7. MathBitsNotebook practice problems
  7. Spot perfect square trinomials - Learning to eyeball patterns like x² + 2dx + d² will speed up your factoring and reduce errors. The more you identify these at a glance, the less time you'll spend calculating (b/2)² every single time. Pattern examples
    8. Pattern examples
  8. Derive the quadratic formula with style - Completing the square is actually how the famous quadratic formula was born! Follow the derivation and you'll see why the formula works and how each piece pops out from the steps you practice. Derivation walkthrough
    9. Derivation walkthrough
  9. Check out extra explanations - When you need a different perspective, head over to eMathHelp for clear notes and examples that reinforce what you've learned. Sometimes a fresh voice is all it takes to click. eMathHelp notes
    10. eMathHelp notes
  10. Watch video tutorials - Visual learners, rejoice! Video lessons can show every algebraic move step-by-step, making it easier to rewind, pause and see how the (b/2)² piece fits in. OnlineMathLearning video
    11. OnlineMathLearning video
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