solve the quadratic equation by completing the squares. Leave in surd form

3x^2+6x-5=0

To solve the quadratic equation \(3x^2 + 6x - 5 = 0\) by completing the square, follow these steps:

### 1. Divide the equation by the coefficient of \(x^2\)
Since the coefficient of \(x^2\) is 3, divide the entire equation by 3:

\[
x^2 + 2x - \frac{5}{3} = 0
\]

### 2. Move the constant term to the right side
Add \(\frac{5}{3}\) to both sides of the equation:

\[
x^2 + 2x = \frac{5}{3}
\]

### 3. Complete the square
To complete the square, take half of the coefficient of \(x\) (which is 2), square it, and add it to both sides. Half of 2 is 1, and \(1^2 = 1\):

\[
x^2 + 2x + 1 = \frac{5}{3} + 1
\]

Simplify the right side:

\[
x^2 + 2x + 1 = \frac{5}{3} + \frac{3}{3} = \frac{8}{3}
\]

### 4. Write the left side as a perfect square
The left side is now a perfect square trinomial:

\[
(x + 1)^2 = \frac{8}{3}
\]

### 5. Solve for \(x\) by taking the square root
Take the square root of both sides:

\[
x + 1 = \pm \sqrt{\frac{8}{3}}
\]

Simplify the square root (note that \(\sqrt{8} = 2\sqrt{2}\)):

\[
x + 1 = \pm \frac{2\sqrt{2}}{\sqrt{3}}
\]

Rationalize the denominator:

\[
x + 1 = \pm \frac{2\sqrt{6}}{3}
\]

### 6. Solve for \(x\)
Finally, subtract 1 from both sides to solve for \(x\):

\[
x = -1 \pm \frac{2\sqrt{6}}{3}
\]

So the solutions are:

\[
x = -1 + \frac{2\sqrt{6}}{3} \quad \text{or} \quad x = -1 - \frac{2\sqrt{6}}{3}
\]

### Final Answer:
\[
x = -1 \pm \frac{2\sqrt{6}}{3}
\]

---

Would you like any further details on this process or have any questions? Here are some related questions you might find useful:

1. How do you solve quadratic equations by factoring?
2. What is the quadratic formula, and how is it derived?
3. How can completing the square be used to derive the quadratic formula?
4. What are the applications of solving quadratic equations in real-world problems?
5. How do you solve quadratic equations using the method of substitution?

**Tip:** When solving quadratic equations by completing the square, always ensure the coefficient of \(x^2\) is 1 before starting the process.

Solve the quadratic equation by completing the squares. Leave in surd form: 3x^2 + 6x - 5 = 0

Learn how to solve the quadratic equation 3x^2 + 6x - 5 by completing the square and leave the solution in surd form. This step-by-step solution covers key algebraic concepts, including completing the square and simplifying surds. Suitable for students in Grades 9-11.

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