Quadratic formula

The quadratic formula provides the roots of a second-degree polynomial equation in the form ax² + bx + c = 0. It is derived from the algebraic process of completing the square and describes the specific values of the independent variable where the parabola intersects the horizontal axis.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

When to use

Use this formula when dealing with a quadratic equation that cannot be easily factored by inspection. It is applicable whenever the equation is expressed in standard form and the leading coefficient 'a' is not zero.

Why it matters

This formula is fundamental in physics for calculating projectile motion and in engineering for optimizing structural designs. It allows for the identification of real or complex solutions, providing essential insight into the behavior of parabolic systems.

Tips

• Calculate the discriminant (b² - 4ac) first to determine the nature of the roots.
• Ensure the equation equals zero before identifying the coefficients a, b, and c.
• Carefully maintain the signs of the coefficients during substitution into the formula.
• Remember that the ± symbol indicates the potential for two distinct solutions.

Common mistakes

• Wrong sign for -b.
• Computing b²-4ac incorrectly.

Real-world example

Finding the time a projectile hits the ground.

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