Questions: How many real solutions does the following quadratic equation have? 4 x^2 + x + 3 = 0 one real solution two real solutions three real solutions no real solutions

Solution Steps

To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant method. The discriminant (\(\Delta\)) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). The number of real solutions depends on the value of the discriminant:

• If \(\Delta > 0\), there are two real solutions.
• If \(\Delta = 0\), there is one real solution.
• If \(\Delta < 0\), there are no real solutions.

For the quadratic equation \(4x^2 + x + 3 = 0\), we identify the coefficients as \(a = 4\), \(b = 1\), and \(c = 3\). The discriminant \(\Delta\) is calculated using the formula: \[ \Delta = b^2 - 4ac \] Substituting the values, we have: \[ \Delta = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47 \]

The value of the discriminant is \(\Delta = -47\). Since \(\Delta < 0\), this indicates that the quadratic equation has no real solutions.

Final Answer