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Quadratic Equation

A quadratic equation is a polynomial equation of degree two in one variable. Its standard form is ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero.

Key Concepts

The highest power of the unknown variable x must be exactly 2 for the equation to be quadratic.

A quadratic equation always has exactly two roots or solutions, which can be real and distinct, real and equal, or not real.

The roots can be calculated using various methods such as factorization, completing the square, or applying the quadratic formula.

Formula & Equation

x = (-b ± √(b^2 - 4ac)) / 2a

x represents the unknown roots, a is the coefficient of x^2, b is the coefficient of x, and c is the constant term. The expression b^2 - 4ac is known as the discriminant.

Common Misconceptions

Myth: The coefficient a in the standard form ax^2 + bx + c = 0 can be zero.

Fact: If a is zero, the x^2 term disappears and the equation becomes a linear equation (bx + c = 0). Therefore, a must always be a non-zero real number.

Real World Applications

Calculating the trajectory of a thrown ball or a fired projectile, which follows a parabolic path.

Determining the dimensions of a rectangular plot of land when the total area and the relationship between its length and width are known.

Frequently Asked Questions

How do you find the roots of a quadratic equation?

You can find the roots by factoring the polynomial, completing the square, or directly substituting the coefficients into the quadratic formula.

What does the discriminant tell us about a quadratic equation?

The discriminant determines the nature of the roots. A positive value indicates two distinct real roots, zero indicates two equal real roots, and a negative value indicates no real roots.

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