Problem 10 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Product of real roots of the equation \(t^{2} x^{2}+\) \(|x|+9=0\), (a) is always positive (b) is always negative (c) does not exist (d) none of these

Step-by-Step Solution

Answer

The product of real roots does not exist.

Step 1: Analyze the Equation

The given equation is \(t^{2}x^{2} + |x| + 9 = 0\). This is a quadratic equation in terms of \(x\). For the quadratic equation to have real roots, the discriminant must be non-negative.

Step 2: Isolate Quadratic Terms

Rewrite the quadratic equation as \(t^{2}x^{2} + |x| + 9 = 0\). Since \(|x|\) is always non-negative and 9 is positive, it suggests that \(t^{2}x^{2}\) has to be non-positive to fulfill the equation.

Step 3: Ensure Non-Negative Discriminant

To have real roots for the equation \(t^{2}x^{2} + |x| + 9 = 0\), the discriminant of the equation \(|x| - 4t^2x^2 - 9\) must be tested for non-negativity. However, observe that a positive quadratic term like \(t^2x^2\) paired with positive constants \(|x| + 9\) suggests a non-real solution even without checking the discriminant, as there's insufficient magnitude to counterbalance these positive terms.

Step 4: Conclude About Real Roots

Since \(t^2x^2\) and \(|x|+9\) combine to prevent the equation from equating to zero for any real \(x\), the equation has no real solutions.

Key Concepts

Understanding the Discriminant in Quadratic Equations

The discriminant is a crucial part of determining the nature of the roots in a quadratic equation, typically expressed as \(ax^2 + bx + c = 0\). The discriminant is given by the formula \(b^2 - 4ac\). This expression helps us decide the nature of the roots without solving the equation fully.

Here's how it works:

• If the discriminant \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, sometimes called a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation has no real roots, meaning the solutions are complex numbers.

By evaluating the discriminant, we can quickly figure out the type of roots we have. In the case of the equation given in the exercise, the discriminant analysis points to no real roots, largely due to the positive terms overpowering any potential for real solutions.

What Are Real Roots?

Real roots are the solutions to an equation that are real numbers. They are the values of \(x\) that satisfy the equation when solved. In a quadratic equation, finding real roots means that when the polynomial is set to zero, the resulting values are not complex numbers.

For an equation to have real roots:

• The discriminant must be zero or positive; negative discriminants result in non-real roots.
• The equation must structurally allow the roots, meaning the interaction of the terms should permit a real solution.

In the exercise, because the terms \(t^{2}x^{2} + |x| + 9\) cannot satisfy the equation with real values of \(x\), the quadratic does not have real roots. The influences of non-negative and positive terms in the equation hinder any real solution.

The Role of Absolute Value in Quadratic Equations

The absolute value, denoted as \(|x|\), is the non-negative value of \(x\) without regard to its sign. In equations, it changes how we handle terms, since \(|x|\) will always yield a non-negative result no matter whether \(x\) is positive or negative.

In quadratic equations:

• Absolute values can complicate the equation since they introduce non-linearity.
• They can act as a constraint in achieving real roots because \(|x|\) is always non-negative, limiting the types of solutions possible.

In this exercise, the incorporated \(|x|\) affects the possibilities of finding real roots. Given that \(|x|\) is positive and added to a positive constant, the equation has no configuration where a non-positive real root balance is achievable, confirming the absence of real roots for the discussed quadratic equation.