Question

Explain why the quadratic formula gives solutions only if \(a \neq 0\) and \(b^{2}-4 a c \geq 0\).

Short answer

The quadratic formula gives real solutions only if \(a \neq 0\) to avoid division by zero and retain the quadratic nature of the equation, and \(b^2 - 4ac \geq 0\) to ensure the square root, or discriminant, is real.

Step-by-step solution

Understand the Quadratic Formula

The quadratic formula is used to solve the quadratic equation of the form \(ax^2 + bx + c = 0\). The solutions of this equation are given by the quadratic formula as: \(x = \[-b \pm \sqrt{b^2 - 4ac}\] / 2a\). The question asks about the conditions \(a \neq 0\) and \(b^2 - 4ac \geq 0\)

Analyze the condition \(a \neq 0\)

The coefficient \(a\) of \(x^2\) is in the denominator of the quadratic formula. Mathematically, division by zero is undefined. Thus, \(a\) cannot be zero for this formula to make sense. If \(a = 0\), the equation would not be a quadratic one, but a linear equation instead.

Analyze the condition \(b^2 - 4ac \geq 0\)

The term under the square root in the quadratic formula, \(b^2 - 4ac\), is called the discriminant. It determines the nature of the roots. If the discriminant is negative, the square root of a negative number becomes an imaginary number. Thus, for real-valued solutions, the discriminant has to be greater than or equal to zero, i.e \(b^2 - 4ac \geq 0\) is required.