Converting a quadratic equation from standard form (ax² + bx + c) to vertex form (a(x - h)² + k) involves completing the square. Here's how to do it:
Steps to Convert Quadratic Form to Vertex Form
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Start with the standard form: y = ax² + bx + c
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Factor out 'a' from the x² and x terms: y = a(x² + (b/a)x) + c. Note: If a = 1, you can skip this step.
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Complete the square: Take half of the coefficient of the x term (b/a), square it ((b/2a)²), and add and subtract it inside the parentheses. This maintains the equation's balance.
- y = a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c
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Rewrite as a perfect square trinomial: The terms inside the parentheses now form a perfect square trinomial, which can be factored.
- y = a((x + (b/2a))² - (b/2a)²) + c
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Distribute 'a' to the subtracted term: Distribute 'a' to the -(b/2a)² term inside the parentheses.
- y = a(x + (b/2a))² - a(b/2a)² + c
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Simplify and rearrange: Simplify the constant terms and rearrange the equation to vertex form (y = a(x - h)² + k). Remember that h = -(b/2a) and k = c - a(b/2a)².
- y = a(x - (-b/2a))² + (c - a(b²/4a²))
- y = a(x - (-b/2a))² + (c - b²/4a)
Example
Convert y = 2x² + 8x + 5 to vertex form:
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Start: y = 2x² + 8x + 5
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Factor out 'a': y = 2(x² + 4x) + 5
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Complete the square: Half of 4 is 2, and 2² is 4. Add and subtract 4 inside the parentheses.
- y = 2(x² + 4x + 4 - 4) + 5
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Rewrite as a perfect square trinomial:
- y = 2((x + 2)² - 4) + 5
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Distribute 'a':
- y = 2(x + 2)² - 2(4) + 5
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Simplify and rearrange:
- y = 2(x + 2)² - 8 + 5
- y = 2(x + 2)² - 3
Therefore, the vertex form is y = 2(x + 2)² - 3. The vertex of the parabola is (-2, -3).
In summary, converting from standard form to vertex form involves strategically completing the square to rewrite the quadratic equation in a format that readily reveals the vertex (h, k) of the parabola.
