express the equation y=3x^2-12x+16 in the form of y=a(x+b)^2+c, where a, b and c are constants

To express the equation \( y = 3x^2 - 12x + 16 \) in the form \( y = a(x + b)^2 + c \), we will complete the square.

### Step 1: Factor out \( a \) from the first two terms.
The equation is \( y = 3x^2 - 12x + 16 \).

Factor out the \( 3 \) from the first two terms:
\[
y = 3(x^2 - 4x) + 16
\]

### Step 2: Complete the square inside the parentheses.
To complete the square, we need to find the value that makes \( x^2 - 4x \) a perfect square trinomial. This value is found by taking half of the coefficient of \( x \), squaring it:
\[
\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4
\]

Now add and subtract 4 inside the parentheses:
\[
y = 3(x^2 - 4x + 4 - 4) + 16
\]

### Step 3: Rewrite as a perfect square trinomial.
Now, \( x^2 - 4x + 4 \) is a perfect square, which can be factored as \( (x - 2)^2 \):
\[
y = 3((x - 2)^2 - 4) + 16
\]

### Step 4: Simplify the equation.
Distribute the 3:
\[
y = 3(x - 2)^2 - 12 + 16
\]
Now simplify the constants:
\[
y = 3(x - 2)^2 + 4
\]

### Final Answer:
The equation in the form \( y = a(x + b)^2 + c \) is:
\[
y = 3(x - 2)^2 + 4
\]
where \( a = 3 \), \( b = -2 \), and \( c = 4 \).

Would you like further details or clarifications?

---

Here are 5 related questions:
1. How do you complete the square for higher-degree polynomials?
2. What are the applications of completing the square in solving quadratic equations?
3. How does the vertex form of a quadratic equation help in graphing?
4. Can all quadratic equations be written in vertex form?
5. How do you convert from vertex form back to standard form?

**Tip**: Completing the square is a key step when deriving the quadratic formula.

Express the equation y=3x^2-12x+16 in the form of y=a(x+b)^2+c, where a, b and c are constants.

Learn how to convert the quadratic equation y=3x^2-12x+16 into vertex form by completing the square. This step-by-step solution covers key algebraic techniques, including factoring and rewriting expressions. Ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation