Math Problem Statement

Which equation is y = (x + 3)2 + (x + 4)2 rewritten in vertex form? y = 2 (x + seven-halves) squared minus one-fourth y = 2 (x + seven-halves) squared minus one-half y = 2(x + 7)2 – 73 y = (x + 7)2 – 24

Solution

We are given the equation:

y=(x+3)2+(x+4)2y = (x + 3)^2 + (x + 4)^2

First, let's expand each term on the right-hand side.

  1. Expand (x+3)2(x + 3)^2: (x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9
  2. Expand (x+4)2(x + 4)^2: (x+4)2=x2+8x+16(x + 4)^2 = x^2 + 8x + 16

Now, add these two expanded terms together:

y=(x2+6x+9)+(x2+8x+16)y = (x^2 + 6x + 9) + (x^2 + 8x + 16)

Simplifying:

y=2x2+14x+25y = 2x^2 + 14x + 25

Next, let's convert this equation into vertex form. Vertex form is:

y=a(xh)2+ky = a(x - h)^2 + k

We first factor out the 2 from the quadratic and linear terms:

y=2(x2+7x)+25y = 2(x^2 + 7x) + 25

Now, complete the square on x2+7xx^2 + 7x. To complete the square, take half of 7 (which is 72\frac{7}{2}) and square it:

(72)2=494\left(\frac{7}{2}\right)^2 = \frac{49}{4}

Add and subtract 494\frac{49}{4} inside the parentheses:

y=2(x2+7x+494494)+25y = 2\left(x^2 + 7x + \frac{49}{4} - \frac{49}{4}\right) + 25

Simplify the expression inside the parentheses:

y=2((x+72)2494)+25y = 2\left(\left(x + \frac{7}{2}\right)^2 - \frac{49}{4}\right) + 25

Now distribute the 2:

y=2(x+72)2492+25y = 2\left(x + \frac{7}{2}\right)^2 - \frac{49}{2} + 25

Convert 25 into a fraction:

25=50225 = \frac{50}{2}

Finally, simplify:

y=2(x+72)2+(492+502)y = 2\left(x + \frac{7}{2}\right)^2 + \left(-\frac{49}{2} + \frac{50}{2}\right)

y=2(x+72)2+12y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}

Thus, the equation in vertex form is:

y=2(x+72)2+12y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}

However, none of the answer choices match exactly. The closest match is:

y=2(x+7/2)21/2y = 2(x + 7/2)^2 - 1/2

Would you like further details or have any questions?


Related Questions:

  1. How do you complete the square on a quadratic equation?
  2. What is the significance of vertex form in quadratic functions?
  3. How do you find the vertex of a quadratic equation in standard form?
  4. Can you explain why we factor out the 2 when completing the square?
  5. How does the transformation of y=x2y = x^2 change when moving to y=a(xh)2+ky = a(x - h)^2 + k?

Tip: Always double-check your arithmetic when completing the square, especially with fractions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vertex Form
Completing the Square

Formulas

Quadratic equation standard form: ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Completing the square: (b/2)^2

Theorems

Completing the Square Theorem
Vertex Form of Quadratic Equations

Suitable Grade Level

Grades 9-11