Math Problem Statement

Solution

To determine which number must be added to complete the square for the equation x2+9x+15=11x^2 + 9x + 15 = 11, we first isolate the quadratic and linear terms:

x2+9x=1115x^2 + 9x = 11 - 15 x2+9x=4x^2 + 9x = -4

Next, we add the square of half the coefficient of xx to both sides. The coefficient of xx is 9, so half of it is 92\frac{9}{2}, and its square is (92)2=814\left(\frac{9}{2}\right)^2 = \frac{81}{4}.

We add 814\frac{81}{4} to both sides of the equation:

x2+9x+(92)2=4+814x^2 + 9x + \left(\frac{9}{2}\right)^2 = -4 + \frac{81}{4} x2+9x+814=4+814x^2 + 9x + \frac{81}{4} = -4 + \frac{81}{4}

Now, we need to find the number that must be added to complete the square on the left side. This number is 814\frac{81}{4}.

Thus, the answer is:

A. 814\boxed{\text{A. } \frac{81}{4}}

Do you want details or have any questions? Here are five more questions you might ask:

  1. How do you determine the number to complete the square in a quadratic equation?
  2. Why is completing the square useful for solving quadratic equations?
  3. How do you solve a quadratic equation once it is in completed square form?
  4. What is the relationship between completing the square and the vertex form of a quadratic equation?
  5. Can completing the square be used for solving quadratic inequalities?

Tip: When completing the square, remember to add the same value to both sides of the equation to maintain the equality. This step is crucial for correctly solving the equation.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Completing the Square

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12