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A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). It is called "quadratic" because of the term \( ax^2 \), which means the variable \( x \) is squared. Quadratic equations are fundamental in algebra because they appear in numerous real-world situations, from physics to finance.
The key is recognizing its standard form, where:

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

The equation \( x^2 - 3x - 4 = 0 \) is a perfect example, where \( a = 1 \), \( b = -3 \), and \( c = -4 \). Solving such equations can help find the values of \( x \) that satisfy the equation, known as the roots or solutions of the equation. These roots can represent various factors depending on the context, such as the points where a projectile hits the ground when solving problems in physics.

The discriminant is a crucial concept when working with quadratic equations. It's there to give us a hint about the nature of the roots of the equation. The formula for the discriminant \( \Delta \) is given by \( \Delta = b^2 - 4ac \). Though it looks straightforward, it holds significant information.
This number tells us:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), it has exactly one real root, also known as a repeated or double root.
• If \( \Delta < 0 \), the equation has no real roots; instead, it has two complex roots.

For our quadratic equation \( x^2 - 3x - 4 = 0 \), the coefficients are \( a = 1 \), \( b = -3 \), and \( c = -4 \), which we use in the discriminant formula: \((-3)^2 - 4(1)(-4)\).
Evaluating the discriminant gives us \( 25 \), indicating that the equation has two distinct real roots.

Knowing the discriminant helps in understanding the roots' nature, but how do we find these roots? One of the most reliable methods is the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The discriminant \( \Delta = b^2 - 4ac \) is part of this formula under the square root sign. It dictates whether the roots are real or complex and how many roots we have.
For the given quadratic equation \( x^2 - 3x - 4 = 0 \), the discriminant calculated as \( 25 \) shows there are two real and distinct solutions. Plugging the values into the quadratic formula:

• \( b = -3 \)
• \( a = 1 \)
• \( c = -4 \)

We solve for \( x \) to find:\[x = \frac{-(-3) \pm \sqrt{25}}{2(1)}\]This gives us:\\[x = \frac{3 \pm 5}{2}\]So, the solutions are \( x = 4 \) and \( x = -1 \). This process demonstrates how understanding and using the discriminant can guide us through finding the roots efficiently.