Complete the square to find the roots of the parabola $^{f(x)=3x^2+11+24x,}$ leaving your answers as surds where necessary.

We are given the quadratic equation $ f(x) = 3x^2 + 24x + 11 $ and asked to find the roots by completing the square. Let's go step by step.

### Step 1: Write the equation in standard quadratic form
The equation is already in the form:

\[
f(x) = 3x^2 + 24x + 11
\]

First, we will factor out the coefficient of $x^2$ (which is 3) from the first two terms:

\[
f(x) = 3(x^2 + 8x) + 11
\]

### Step 2: Complete the square
Now, we complete the square inside the parentheses. To do that, we take the coefficient of $x$ (which is 8), divide it by 2, and then square it:

\[
\left(\frac{8}{2}\right)^2 = 16
\]

We add and subtract this value inside the parentheses:

\[
f(x) = 3(x^2 + 8x + 16 - 16) + 11
\]
\[
f(x) = 3((x + 4)^2 - 16) + 11
\]

### Step 3: Simplify the equation
Now expand and simplify:

\[
f(x) = 3(x + 4)^2 - 48 + 11
\]
\[
f(x) = 3(x + 4)^2 - 37
\]

### Step 4: Solve for the roots
To find the roots, we set $f(x) = 0$:

\[
3(x + 4)^2 - 37 = 0
\]
\[
3(x + 4)^2 = 37
\]
\[
(x + 4)^2 = \frac{37}{3}
\]

### Step 5: Take the square root of both sides
Now, take the square root of both sides:

\[
x + 4 = \pm \sqrt{\frac{37}{3}}
\]

Simplify the square root:

\[
x + 4 = \pm \frac{\sqrt{37}}{\sqrt{3}} = \pm \frac{\sqrt{37}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \pm \frac{\sqrt{111}}{3}
\]

### Step 6: Solve for $x$
Finally, subtract 4 from both sides:

\[
x = -4 \pm \frac{\sqrt{111}}{3}
\]

### Final Answer:
The roots of the quadratic equation are:

\[
x = -4 \pm \frac{\sqrt{111}}{3}
\]

Would you like me to explain any specific steps or provide more details?

Here are some related questions for further exploration:
1. How do you apply the quadratic formula to solve the same equation?
2. How does completing the square compare with factoring?
3. Can we always complete the square for any quadratic equation?
4. What is the geometric meaning of completing the square for a parabola?
5. How does the discriminant help in determining the nature of the roots?

**Tip:** Completing the square is particularly useful when solving quadratic equations that do not factor easily.

Complete the square to find the roots of the parabola f(x)=3x^2+11+24x, leaving your answers as surds where necessary.

Learn how to complete the square to solve the quadratic equation f(x) = 3x^2 + 24x + 11. This step-by-step guide covers key concepts like completing the square and simplifying surds, perfect for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation