Questions and Answers
Which of the following expressions are perfect-square trinomials? (Select all that apply)
Complete the expression so it forms a perfect-square trinomial: x² - 5x + ____
25/4
A student says that if 5x² = 20, then x must be equal to 2. Do you agree or disagree with the student? Justify your answer.
The answer to the problem would be 2 and -2.
Given (x - 7)² = 36, select the values of x.
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Given (x - 1)² = 50, select the values of x.
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What method would you choose to solve the equation 2x² - 7 = 9? Explain why you chose this method.
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Solve x² - 16x + ____ = -12 by completing the steps. First, subtract from each side of the equation.
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Next, add ____ to each side of the equation to complete the square.
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Now, write x² - 16x + 64 = -8 as ____.
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Take the square root of both sides to get the solutions.
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Complete the square: x² - 6x + ____ = -13 + ____
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Factor the trinomial and simplify: (x + ____ )² = ____.
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The solution to x² - 10x = 24 is ____.
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The solution to 2x² - 11 = 87 is ____.
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The solution to 3x² - 12x + 24 = 0 is ____.
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Complete the square to write 16t² - 96t + 48 = 0 as ____.
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Solve (t - 3)² = 6. The arrow is at a height of 48 ft after approximately ____.
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Study Notes
Perfect-Square Trinomials
- Perfect-square trinomials include expressions that can be factored into the form (a + b)².
- From the examples provided, the following are perfect-square trinomials:
- 4x² + 12x + 9
- x² + 20x + 100
- x² + 4x + 16
Completing the Square
- Completing the square involves transforming a quadratic expression into a perfect-square trinomial.
- For x² - 5x, adding (25/4) completes the square.
Equation Solutions
- If 5x² = 20, the solutions are x = 2 and x = -2, showcasing the principle of finding values for x in quadratic equations.
- Given (x - 7)² = 36 leads to solutions x = 13 and x = 1 through taking the square root of both sides.
- For (x - 1)² = 50, the valid solutions are x = 51, reaffirming the value of checking multiple potential solutions.
Equation Solving Processes
- To solve the equation 2x² - 7 = 9:
- Divide all terms by two, simplifying the equation.
- After rearranging, the solution involves completing the square.
Completing Steps in Quadratics
- To solve x² - 16x = -12 by completing the square:
- First, determine the value needed to balance the equation, which is 60.
- Then, add 64 to both sides, allowing the equation to be expressed in a perfect-square form.
Final Steps in Solutions
- For the expression x² - 16x + 64 = -8, solving helps identify roots corresponding to the square root of both sides, yielding solutions of x = 8 and x = 2.
Factoring and Simplifying
- The trinomial can be factored and simplified to the form (x + b)², where specific values like -3 and -4 play a crucial role.
Quadratic Solutions
- Solutions for various quadratic equations include:
- For x² - 10x = 24, the result is indicated as solution C.
- For 2x² - 11 = 87, the solution is acknowledged as A.
- The solution to 3x² - 12x + 24 = 0 is denoted as B.
Contextual Application
- Completing the square can also transform equations such as 16t² - 96t + 48 = 0 into simpler forms, helping clearly understand the roots.
- Solving (t - 3)² = 6 yields height problems leading to solutions around 5 and 5.45 for applications like projectile motion.
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Description
Test your understanding of perfect-square trinomials and completing the square with these flashcards. This quiz covers various expressions and their properties, encouraging you to engage with key concepts in quadratic equations.
