Questions: Compute the discriminant. Then determine the number and type of solutions of the given equation. x^2 - 7x - 4 = 0 What is the discriminant? Choose the sentence that describes the number and type of solutions of the quadratic equation. A. There are two unequal real solutions. B. There is one real solution. C. There are an infinite number of real solutions. D. There are two imaginary solutions.

Solution Steps

The given quadratic equation is \(x^2 - 7x - 4 = 0\). From this equation, we identify the coefficients:

• \(a = 1\)
• \(b = -7\)
• \(c = -4\)

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \[ \Delta = b^2 - 4ac \] Substituting the values of \(a\), \(b\), and \(c\) into the formula, we get: \[ \Delta = (-7)^2 - 4 \times 1 \times (-4) = 49 + 16 = 65 \]

The discriminant \(\Delta = 65\) is positive. According to the properties of the discriminant:

• If \(\Delta > 0\), there are two distinct real solutions.

Final Answer