Solve for x using the quadratic formula in the equation: $$ 3x^{2} - 6x + 2 = 0 $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation $$ 3x^{2} - 6x + 2 = 0 $$ using the quadratic formula, follow these steps:

### **Quadratic Formula Recap**

For any quadratic equation of the form $$ ax^{2} + bx + c = 0 $$, the solutions for $$ x $$ are given by:

$$
x = \frac{ -b \pm \sqrt{b^{2} - 4ac} }{2a}
$$

### **Given Equation Analysis**

Identify the coefficients from the equation $$ 3x^{2} - 6x + 2 = 0 $$:

- $$ a = 3 $$
- $$ b = -6 $$
- $$ c = 2 $$

### **Plugging into the Formula**

Substitute the values of $$ a $$, $$ b $$, and $$ c $$ into the quadratic formula:

$$
x = \frac{ -(-6) \pm \sqrt{(-6)^{2} - 4 \times 3 \times 2} }{2 \times 3}
$$

### **Simplifying Step-by-Step**

1. **Calculate the Discriminant ($$ \Delta $$)**:

$$
\Delta = b^{2} - 4ac = (-6)^{2} - 4 \times 3 \times 2 = 36 - 24 = 12
$$

2. **Compute the Square Root of the Discriminant**:

$$
\sqrt{\Delta} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
$$

3. **Substitute Back into the Formula**:

$$
x = \frac{6 \pm 2\sqrt{3}}{6}
$$

4. **Simplify the Fraction**:

$$
x = \frac{6}{6} \pm \frac{2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}
$$

### **Final Solutions**

$$
x = 1 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x = 1 - \frac{\sqrt{3}}{3}
$$

### **Approximate Decimal Values**

For a numerical approximation:

- $$ \sqrt{3} \approx 1.732 $$
- $$ \frac{\sqrt{3}}{3} \approx 0.577 $$

Thus, the solutions are approximately:

$$
x \approx 1 + 0.577 = 1.577 \quad \text{and} \quad x \approx 1 - 0.577 = 0.423
$$

### **Conclusion**

The quadratic equation $$ 3x^{2} - 6x + 2 = 0 $$ has two real solutions:

$$
x = 1 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x = 1 - \frac{\sqrt{3}}{3}
$$

Or approximately:

$$
x \approx 1.577 \quad \text{and} \quad x \approx 0.423
$$
The solutions to the equation $$ 3x^{2} - 6x + 2 = 0 $$ are: $$ x = 1 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x = 1 - \frac{\sqrt{3}}{3} $$ Approximately: $$ x \approx 1.577 \quad \text{and} \quad x \approx 0.423 $$

Solve for x using the quadratic formula in the equation:

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions