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Questions and Answers
What are the solutions to the equation $x^2 + 12x + 35 = 0$?
What are the solutions to the equation $x^2 + 12x + 35 = 0$?
- x = -3; x = -4
- x = -5; x = -7 (correct)
- x = -11; x = 7
- x = 5; x = -6
What are the solutions to the equation $x^2 + 4x - 77 = 0$?
What are the solutions to the equation $x^2 + 4x - 77 = 0$?
- x = -11; x = 7 (correct)
- x = -2; x = 3
- x = 1/3; x = -1.5
- x = 5; x = -6
What are the solutions to the equation $x^2 + 7x + 12 = 0$?
What are the solutions to the equation $x^2 + 7x + 12 = 0$?
- x = 0; x = 7
- x = 1; x = -9
- x = 3; x = 5
- x = -3; x = -4 (correct)
What are the solutions to the equation $x^2 + x - 30 = 0$?
What are the solutions to the equation $x^2 + x - 30 = 0$?
What are the solutions to the equation $2x^2 - 2x - 12 = 0$?
What are the solutions to the equation $2x^2 - 2x - 12 = 0$?
What are the solutions to the equation $x^2 - 16x - 17 = 0$?
What are the solutions to the equation $x^2 - 16x - 17 = 0$?
What is the solution to the equation $x^2 - 16x + 64 = 0$?
What is the solution to the equation $x^2 - 16x + 64 = 0$?
What is the solution to the equation $x^2 + 14x + 49 = 0$?
What is the solution to the equation $x^2 + 14x + 49 = 0$?
What are the solutions to the equation $x^2 - 64 = 0$?
What are the solutions to the equation $x^2 - 64 = 0$?
What are the solutions to the equation $x^2 + 4x - 60 = 0$?
What are the solutions to the equation $x^2 + 4x - 60 = 0$?
What are the solutions to the equation $6x^2 + 7x - 3 = 0$?
What are the solutions to the equation $6x^2 + 7x - 3 = 0$?
What are the solutions to the equation $5x^2 - 45 = 0$?
What are the solutions to the equation $5x^2 - 45 = 0$?
What are the solutions to the equation $x^2 - 3x - 4 = 0$?
What are the solutions to the equation $x^2 - 3x - 4 = 0$?
What are the solutions to the equation $x^2 - 8x + 15 = 0$?
What are the solutions to the equation $x^2 - 8x + 15 = 0$?
What are the solutions to the equation $x^2 - 7x = 0$?
What are the solutions to the equation $x^2 - 7x = 0$?
What are the solutions to the equation $(x + 4)^2 = 25$?
What are the solutions to the equation $(x + 4)^2 = 25$?
What are the solutions to the equation $(x - 3)^2 = 64$?
What are the solutions to the equation $(x - 3)^2 = 64$?
What are the solutions to the equation $5x^2 - 10x - 120 = 0$?
What are the solutions to the equation $5x^2 - 10x - 120 = 0$?
What are the solutions to the equation $7x^2 + 14x - 21 = 0$?
What are the solutions to the equation $7x^2 + 14x - 21 = 0$?
What are the solutions to the equation $x^2 - 17 = 8$?
What are the solutions to the equation $x^2 - 17 = 8$?
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Study Notes
Key Methods for Solving Quadratic Equations
-
Factoring: A common method where the equation is expressed as a product of binomials.
- Example: x² + 12x + 35 = 0 yields solutions x = -5 and x = -7.
- x² + 7x + 12 = 0 yields x = -3 and x = -4.
- x² + 4x - 60 = 0 yields x = 6 and x = -10.
- x² - 3x - 4 = 0 yields x = -1 and x = 4.
-
Completing the Square: Transforming the quadratic into a perfect square trinomial.
- Example: x² + 4x - 77 = 0 results in x = -11 and x = 7.
- 2x² - 2x - 12 = 0 leads to x = -2 and x = 3.
- x² - 8x + 15 = 0 gives x = 3 and x = 5.
- 7x² + 14x - 21 = 0 provides roots x = 1 and x = -3.
-
Quadratic Formula: A formula that provides the roots of any quadratic equation of the form ax² + bx + c = 0.
- Example: x² + x - 30 = 0 results in x = 5 and x = -6.
- 6x² + 7x - 3 = 0 yields x = 1/3 and x = -1.5.
- 5x² - 10x - 120 = 0 results in x = 6 and x = -4.
- x² - 7x = 0 gives the solutions x = 0 and x = 7.
-
Using Square Roots: Applies when the quadratic can be rewritten in the form (x ± a)² = b.
- Example: x² - 64 = 0 leads to x = 8 and x = -8.
- (x + 4)² = 25 results in x = 1 and x = -9.
- (x - 3)² = 64 provides roots x = 11 and x = -5.
- x² - 17 = 8 results in x = -5 and x = 5.
Summary
- Various methods exist to solve quadratic equations: factoring, completing the square, quadratic formula, and using square roots.
- Each method is effective depending on the specific form of the quadratic equation.
- Solutions can yield rational or irrational numbers, impacting the approach chosen for solving.
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