Math Problem Statement

f(x) = 3x^2 + 21x - 36. Turn into f(x) = a(x - h)^2 + k.

Solution

You are looking to convert the quadratic function f(x)=3x2+21x36f(x) = 3x^2 + 21x - 36 into the form f(x)=a(xh)2+kf(x) = a(x - h)^2 + k, which is called the vertex form. To do this, we need to complete the square. Let me walk you through the steps:

Step 1: Factor out the coefficient of x2x^2 from the first two terms.

We have: f(x)=3(x2+7x)36f(x) = 3(x^2 + 7x) - 36

Step 2: Complete the square inside the parentheses.

To complete the square, take half of the coefficient of xx (which is 7), square it, and add and subtract it inside the parentheses: f(x)=3(x2+7x+(72)2(72)2)36f(x) = 3 \left( x^2 + 7x + \left( \frac{7}{2} \right)^2 - \left( \frac{7}{2} \right)^2 \right) - 36 f(x)=3(x2+7x+494494)36f(x) = 3 \left( x^2 + 7x + \frac{49}{4} - \frac{49}{4} \right) - 36 f(x)=3((x+72)2494)36f(x) = 3 \left( \left( x + \frac{7}{2} \right)^2 - \frac{49}{4} \right) - 36

Step 3: Simplify the expression.

Now, distribute the factor of 3: f(x)=3(x+72)23×49436f(x) = 3 \left( x + \frac{7}{2} \right)^2 - 3 \times \frac{49}{4} - 36 f(x)=3(x+72)2147436f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{147}{4} - 36

Convert -36 to have a denominator of 4: f(x)=3(x+72)214741444f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{147}{4} - \frac{144}{4} f(x)=3(x+72)22914f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{291}{4}

Final Answer:

The quadratic function in vertex form is: f(x)=3(x+72)22914f(x) = 3 \left( x + \frac{7}{2} \right)^2 - \frac{291}{4}

This is the desired form f(x)=a(xh)2+kf(x) = a(x - h)^2 + k, where:

  • a=3a = 3,
  • h=72h = -\frac{7}{2}, and
  • k=2914k = -\frac{291}{4}.

Would you like more details or have any questions?

Here are five related questions you could explore:

  1. How do you derive the vertex from the vertex form of a quadratic equation?
  2. What is the importance of completing the square in solving quadratic equations?
  3. How does the coefficient aa affect the shape of the parabola?
  4. What is the axis of symmetry for the given function?
  5. How would you find the x-intercepts of this quadratic function?

Tip: When completing the square, always ensure that you maintain balance in the equation by adjusting constants both inside and outside the parentheses correctly.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Completing the Square
Algebra

Formulas

Standard form of a quadratic function: ax^2 + bx + c
Vertex form of a quadratic function: a(x - h)^2 + k

Theorems

Method of Completing the Square

Suitable Grade Level

Grades 9-11