The roots of the equation `5x^(2)+10x-7=0` are

Options

A. Positive
B. Negative
C. of oppsote signs
D. rational

Answer

To find the roots of the quadratic equation \(5x^2 + 10x - 7 = 0\), we will follow these steps:

### Step 1: Identify coefficients
We start by identifying the coefficients from the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

Here,
- \(a = 5\)
- \(b = 10\)
- \(c = -7\)

### Step 2: Calculate the discriminant
Next, we calculate the discriminant \(D\) using the formula:
\[ D = b^2 - 4ac \]

Substituting the values:
\[ D = 10^2 - 4 \cdot 5 \cdot (-7) \]
\[ D = 100 + 140 = 240 \]

### Step 3: Determine the nature of the roots
Since \(D > 0\), this indicates that the quadratic equation has two distinct real roots.

### Step 4: Calculate the roots using the quadratic formula
We use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

Substituting the values:
\[ x = \frac{-10 \pm \sqrt{240}}{2 \cdot 5} \]
\[ x = \frac{-10 \pm \sqrt{240}}{10} \]
\[ x = -1 \pm \frac{\sqrt{240}}{10} \]

### Step 5: Simplify \(\sqrt{240}\)
Next, we simplify \(\sqrt{240}\):
\[ \sqrt{240} = \sqrt{16 \cdot 15} = 4\sqrt{15} \]

So we have:
\[ x = -1 \pm \frac{4\sqrt{15}}{10} \]
\[ x = -1 \pm \frac{2\sqrt{15}}{5} \]

### Step 6: Write the final roots
Thus, the roots of the equation are:
\[ x_1 = -1 + \frac{2\sqrt{15}}{5} \]
\[ x_2 = -1 - \frac{2\sqrt{15}}{5} \]

### Summary of the roots
The roots of the equation \(5x^2 + 10x - 7 = 0\) are:
\[ x_1 = -1 + \frac{2\sqrt{15}}{5}, \quad x_2 = -1 - \frac{2\sqrt{15}}{5} \]

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