Problems Based on Discriminant

The discriminant is a key concept in the study of quadratic equations. It provides critical information about the nature of the roots of a quadratic equation without actually solving the equation. The discriminant is denoted by the symbol $\Delta$ and is derived from the standard form of a quadratic equation:

$$ ax^2 + bx + c = 0 $$

where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.

The Discriminant Formula

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula:

$$ \Delta = b^2 - 4ac $$

Interpretation of the Discriminant

The value of the discriminant tells us the nature and number of roots of the quadratic equation:

Discriminant ($\Delta$)	Nature of Roots	Number of Real Roots
$\Delta > 0$	Real and distinct	Two
$\Delta = 0$	Real and equal	One (repeated root)
$\Delta < 0$	Complex	No real roots

Examples

Let's look at some examples to understand how the discriminant is used to determine the nature of the roots of a quadratic equation.

Example 1: Real and Distinct Roots

Consider the quadratic equation $2x^2 - 4x + 1 = 0$. To find the discriminant:

$$ \Delta = (-4)^2 - 4(2)(1) = 16 - 8 = 8 $$

Since $\Delta > 0$, the equation has two real and distinct roots.

Example 2: Real and Equal Roots

Consider the quadratic equation $x^2 - 2x + 1 = 0$. To find the discriminant:

$$ \Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0 $$

Since $\Delta = 0$, the equation has one real root (a repeated root).

Example 3: Complex Roots

Consider the quadratic equation $x^2 + x + 1 = 0$. To find the discriminant:

$$ \Delta = (1)^2 - 4(1)(1) = 1 - 4 = -3 $$

Since $\Delta < 0$, the equation has no real roots; the roots are complex.

Problems Based on Discriminant

When solving problems based on the discriminant, you may be asked to:

1. Determine the nature of the roots without solving the equation.
2. Find conditions on the coefficients $a$, $b$, and $c$ for the roots to be real, equal, or complex.
3. Modify the equation to ensure a certain type of root.

Problem 1: Nature of Roots Without Solving

Given: $3x^2 - 5x + 2 = 0$

Find: The nature of the roots.

Solution:

$$ \Delta = (-5)^2 - 4(3)(2) = 25 - 24 = 1 $$

Since $\Delta > 0$, the roots are real and distinct.

Problem 2: Conditions for Real and Equal Roots

Given: $ax^2 + bx + c = 0$

Find: The condition for the roots to be real and equal.

Solution:

For the roots to be real and equal, $\Delta$ must be zero:

$$ \Delta = b^2 - 4ac = 0 $$

Thus, the condition is $b^2 = 4ac$.

Problem 3: Modifying the Equation

Given: $x^2 + px + 15 = 0$

Find: The value of $p$ for which the roots are real and distinct.

Solution:

For real and distinct roots, $\Delta$ must be greater than zero:

$$ \Delta = p^2 - 4(1)(15) > 0 $$ $$ p^2 - 60 > 0 $$ $$ p^2 > 60 $$

Thus, $p$ must satisfy $p > \sqrt{60}$ or $p < -\sqrt{60}$.

Conclusion

The discriminant is a powerful tool in the analysis of quadratic equations. By examining the value of $\Delta$, we can quickly determine the nature of the roots without solving the equation. This is particularly useful in algebra, calculus, and other areas of mathematics where the properties of functions are studied.