Question

Prove that the equation \(a x^{2}+b x+c=0\), in which \(a, b\) and \(c\) are real and \(a>0\), has two real distinct solutions IFF \(b^{2}>4 a c\).

Short answer

ax^{2}+bx+c=0,will have two distinct real solutions IFF b^{2} > 4ac.

Step-by-step solution

Understand the Quadratic Equation

The standard form of a quadratic equation is given by x^{2}+b x+c=0i. We need to prove that this equation has two real distinct solutions if and only if (IFF) the discriminant b^{2}-4aci.e. greater than zero.

Recall the Discriminant

The discriminant of a quadratic equation ax^{2}+bx+c=0i is given by b^{2}-4aci. This value determines the nature of the roots of the quadratic equation.

Analyze the Discriminant Conditions

The quadratic equation will have two distinct real solutions if (b^2 - 4ac) > 0i.

Prove the IFF Condition

To have two real distinct solutions: b^{2} - 4aci. Must be greater than zero. i.e. b^{2} > 4acthis is true. If b^{2} <= 4acthen the equation doesn't have two real distinct solutions, QED.

Key concepts

Discriminant

The discriminant is a key concept in solving quadratic equations. It's a part of the quadratic formula used to find the roots of the equation. The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by: \(\Delta = b^2 - 4ac\). The value of the discriminant helps us determine the nature of the roots:

• If \(\Delta > 0\), there are two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution (or a repeated root).
• If \(\Delta < 0\), there are no real solutions; instead, the equation has two complex solutions.

Understanding the role of the discriminant helps us quickly figure out what kind of solutions to expect from the quadratic equation without actually solving it.

Real Solutions

Real solutions refer to the values of \(x\) that make the quadratic equation true in the realm of real numbers. For the quadratic equation \(ax^2 + bx + c = 0\), there are some important points to remember:

• When the discriminant \(\Delta > 0\), the equation has two distinct real solutions. These solutions can be found using the quadratic formula: \[x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}}\].
• When \(\Delta = 0\), the equation has a single real solution, also calculated with the quadratic formula, but since the square root of zero is zero, we get: \[ x = \frac{{-b}}{{2a}} \].
• When \(\Delta < 0\), the equation does not have any real solutions. Instead, it has complex or imaginary solutions.

Knowing whether a quadratic equation has real solutions is important in many fields of study, including physics and engineering.

Proof of Conditions

To prove that the quadratic equation \(ax^2 + bx + c = 0\) has two real distinct solutions if and only if \(b^2 > 4ac\), follow these steps:

• Start with the equation \(ax^2 + bx + c = 0\).
• Recall the discriminant, which is \(\Delta = b^2 - 4ac\).
• For the equation to have two distinct real solutions, the discriminant must be greater than zero, i.e., \(\Delta > 0\).

Therefore, we need \(b^2 - 4ac > 0\) to ensure that the quadratic equation has two real distinct solutions. We conclude that:

• If \(b^2 > 4ac\), the discriminant is positive, indicating two distinct real solutions.
• If \(b^2 \leq 4ac\), the discriminant is non-positive, indicating no or only one real solution.

This logical proof confirms that the condition \(b^2 > 4ac\) is both necessary and sufficient for the existence of two distinct real solutions in a quadratic equation.