Math Problem Statement

Which equation is y = –6x2 + 3x + 2 rewritten in vertex form? y = negative 6 (x minus 1) squared + 8 y = negative 6 (x + one-fourth) squared + thirteen-eighths y = negative 6 (x minus one-fourth) squared + nineteen-eighths y = negative 6 (x minus one-half) squared + seven-halves

Solution

To rewrite the quadratic equation y=6x2+3x+2y = -6x^2 + 3x + 2 in vertex form, we need to complete the square.

Steps to rewrite in vertex form:

  1. Start with the given equation:
    y=6x2+3x+2y = -6x^2 + 3x + 2

  2. Factor out 6-6 from the first two terms:
    y=6(x212x)+2y = -6(x^2 - \frac{1}{2}x) + 2

  3. Complete the square:
    To complete the square, take half of the coefficient of xx inside the parentheses (which is 12-\frac{1}{2}), square it, and add and subtract this value inside the parentheses: (122)2=(14)2=116\left(\frac{-\frac{1}{2}}{2}\right)^2 = \left(-\frac{1}{4}\right)^2 = \frac{1}{16} So we add and subtract 116\frac{1}{16} inside the parentheses: y=6(x212x+116116)+2y = -6\left(x^2 - \frac{1}{2}x + \frac{1}{16} - \frac{1}{16}\right) + 2

  4. Rewrite the perfect square: y=6((x14)2116)+2y = -6\left(\left(x - \frac{1}{4}\right)^2 - \frac{1}{16}\right) + 2

  5. Simplify the equation: Now, distribute the 6-6: y=6(x14)2+616+2y = -6(x - \frac{1}{4})^2 + \frac{6}{16} + 2 y=6(x14)2+38+2y = -6(x - \frac{1}{4})^2 + \frac{3}{8} + 2 y=6(x14)2+38+168y = -6(x - \frac{1}{4})^2 + \frac{3}{8} + \frac{16}{8} y=6(x14)2+198y = -6(x - \frac{1}{4})^2 + \frac{19}{8}

Thus, the vertex form of the equation is: y=6(x14)2+198y = -6\left(x - \frac{1}{4}\right)^2 + \frac{19}{8}

The correct choice is: y=6(x14)2+198\boxed{y = -6\left(x - \frac{1}{4}\right)^2 + \frac{19}{8}}


Would you like further details or have any questions?

Here are five related questions for deeper understanding:

  1. How do you complete the square in a general quadratic equation?
  2. What is the vertex of the quadratic equation in vertex form?
  3. How does the value of aa affect the graph of a quadratic function in vertex form?
  4. Can you convert the vertex form back into standard form?
  5. What are the benefits of expressing a quadratic equation in vertex form?

Tip: The vertex form of a quadratic equation is useful because it easily reveals the vertex (h,k)(h, k) of the parabola, making it easier to graph and analyze.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vertex Form
Completing the Square

Formulas

Standard form of quadratic equation: y = ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Completing the square

Theorems

Quadratic Formula
Properties of Parabolas

Suitable Grade Level

Grades 9-12