Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Recall that a quadratic equation \(a x^{2}+b x+c=0\) has two real solutions if and only if the discriminant \(b^{2}-4 a c\) is positive. Prove that if \(a\) and \(c\) have different signs then the quadratic equation has two real solutions.

Step-by-Step Solution

Answer

If \(a\) and \(c\) have different signs, the discriminant \(b^{2}-4ac\) is positive, ensuring two real solutions.

Step 1 - Understanding the Discriminant

Recall that for a quadratic equation in the form of \(a x^{2}+b x+c=0\), the discriminant is defined as \(D = b^{2}-4ac\). A quadratic equation has two real solutions if \(D > 0\).

Step 2 - Analyzing the Conditions

We need to prove that if the coefficients \(a\) and \(c\) have different signs, the quadratic equation has two real solutions. There are two cases to consider: \(a > 0\) and \(c < 0\), or \(a < 0\) and \(c > 0\).

Step 3 - Case 1: \(a > 0\) and \(c < 0\)

If \(a > 0\) and \(c < 0\), then \(-4ac\) is positive because the product of a positive number and a negative number is negative, so the negative sign makes it positive. Since \(b^{2}\) is always non-negative, adding a positive number \(-4ac\) to \(b^{2}\) ensures that \(D > 0\). Hence, \(b^{2} - 4ac > 0\).

Step 4 - Case 2: \(a 0\)

If \(a < 0\) and \(c > 0\), again \(-4ac\) is positive because the product of a negative number \(a\) and a positive number \(c\) is negative, so \(-4ac\) becomes positive. Since \(b^{2}\) is non-negative, \(b^{2} + (-4ac)\) ensures \(D > 0\), implying \(b^{2} - 4ac > 0\).

Step 5 - Conclusion

In both cases, we have shown that \(b^{2} - 4ac > 0\), thus ensuring that the quadratic equation has two real solutions. Therefore, if the coefficients \(a\) and \(c\) have different signs, the quadratic equation will have two real solutions.

Key Concepts

Understanding the Discriminant

In a quadratic equation of the form \(a x^{2} + b x + c = 0\), the discriminant is a key factor in determining the nature of the equation's solutions. The discriminant, denoted as \(D\), is given by the formula \(D = b^{2} - 4ac\). It plays a crucial role in understanding whether the equation has real or complex solutions. A quadratic equation will have:

• Two real solutions if \(D > 0\)
• One real solution if \(D = 0\)
• Two complex solutions if \(D < 0\)

For our purposes, we focus on the condition \(D > 0\), which ensures that the quadratic equation has two real solutions. This happens because the square root of the discriminant will be a positive real number, leading to two distinct solutions when applying the quadratic formula.

Conditions for Real Solutions

The essence of proving that a quadratic equation has two real solutions lies in demonstrating that its discriminant is positive. We specifically need to examine the scenario where the coefficients \(a\) and \(c\) have different signs. Let's break this down:

• If \(a > 0\) and \(c < 0\): Here, \(-4ac\) is positive because the product of a positive coefficient \(a\) and a negative coefficient \(c\) results in a negative value, and the negative sign in \(-4ac\) converts it to positive.
• If \(a < 0\) and \(c > 0\): Similarly, \(-4ac\) remains positive because the product of a negative coefficient \(a\) and a positive coefficient \(c\) is negative, and multiplying by \(-4\) makes it positive again.

In both cases, since \(b^{2}\) is non-negative, adding any positive number to it will ensure that \(b^{2} - 4ac > 0\), leading to a positive discriminant and thus guaranteeing two real solutions.

Role of Coefficients

Coefficients in a quadratic equation have a significant impact on the nature of its solutions. Take the standard form \(a x^{2} + b x + c = 0\); here, \(a\), \(b\), and \(c\) are the coefficients:

• \(a\) is the coefficient of \(x^2\) (the leading coefficient).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

For a quadratic equation to have two real solutions, the discriminant \(b^{2} - 4ac\) must be positive. The signs of \(a\) and \(c\) play a crucial role in this. Specifically, if \(a\) and \(c\) have opposite signs, then \(-4ac\) becomes positive. This addition of a positive value to \(b^2\)—which is always non-negative—ensures that \(b^2 - 4ac > 0\). This shows that the coefficients \(a\) and \(c\) are key to determining when a quadratic equation has two real solutions.