solve three x squared plus five x minus two with the quadratic formula

To solve the quadratic equation \(3x^2 + 5x - 2 = 0\) using the quadratic formula, we follow these steps:

### Step 1: Identify the coefficients
The quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 3\)
- \(b = 5\)
- \(c = -2\)

### Step 2: Write down the quadratic formula
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 3: Substitute the values of \(a\), \(b\), and \(c\) into the formula
\[
x = \frac{-5 \pm \sqrt{5^2 - 4(3)(-2)}}{2(3)}
\]

### Step 4: Simplify inside the square root
First, calculate \(b^2 - 4ac\):
\[
b^2 - 4ac = 5^2 - 4(3)(-2) = 25 + 24 = 49
\]

So the expression becomes:
\[
x = \frac{-5 \pm \sqrt{49}}{6}
\]

### Step 5: Simplify the square root
\(\sqrt{49} = 7\), so:
\[
x = \frac{-5 \pm 7}{6}
\]

### Step 6: Calculate the two possible values for \(x\)
1. For \(x_1\), using the plus sign:
\[
x_1 = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3}
\]

2. For \(x_2\), using the minus sign:
\[
x_2 = \frac{-5 - 7}{6} = \frac{-12}{6} = -2
\]

### Final Answer:
The solutions to the equation \(3x^2 + 5x - 2 = 0\) are:
\[
x_1 = \frac{1}{3}, \quad x_2 = -2
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Solve \(2x^2 + 4x - 6 = 0\) using the quadratic formula.
2. What are the roots of the equation \(4x^2 - 12x + 9 = 0\)?
3. Find the discriminant of \(5x^2 + 2x + 1 = 0\) and interpret its meaning.
4. Solve \(x^2 - 7x + 10 = 0\) using the quadratic formula.
5. How does the value of the discriminant determine the nature of the roots in a quadratic equation?

**Tip:** The discriminant (\(b^2 - 4ac\)) helps determine the nature of the roots: if it's positive, there are two real roots; if zero, one real root; if negative, two complex roots.

Learn how to solve the quadratic equation 3x^2 + 5x - 2 with a detailed step-by-step solution using the quadratic formula. This problem covers key algebraic concepts, including the quadratic equation formula and the quadratic formula theorem. Suitable for students in Grades 7-9.

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