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Jan 30, 2026

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3 pages

Master Quadratic Equations: Easy Methods Explained

Jenna

@jenna_bmem

Quadratic equations are powerful mathematical tools that appear in countless... Show more

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Solving Quadratics

Def: a quadratic equation is an equation with a degree 2
$x² + Ay² - Bz2$

A quadratic equation with one variable has th

Understanding and Solving Quadratics

A quadratic equation in one variable follows the standard form Ax² + Bx + C = 0, where A, B, and C are constants. When we solve these equations, we're looking for roots (also called zeros) - the values that make the equation equal to zero.

There are three main methods to solve quadratics: factoring, completing the square, and using the quadratic formula. To choose the right method, calculate the discriminant $b² - 4ac$ which reveals important information about your solutions. When the discriminant is positive, you'll have two real solutions; negative means two imaginary solutions; zero indicates exactly one solution.

When factoring quadratics, first set the equation to zero, then find two numbers that both add up to B and multiply to give A×C. This allows you to rewrite the middle term, group terms effectively, and then find your solution by setting each factor equal to zero.

Pro Tip: If the discriminant $b² - 4ac$ is a perfect square, the quadratic is likely factorable - making the factoring method your fastest approach!

Completing the Square and Quadratic Formula

Completing the square transforms quadratics into perfect square form. Start with your equation Ax² + Bx = C, then find B/2 and square it. Add this value to both sides, allowing you to rewrite the left side as a perfect square $x + number$ ². The "number" will always be B/2. Take the square root of both sides to solve for x.

For equations where the coefficient of x² isn't 1, factor out that number first. Then apply the completing the square method to the expression in parentheses, making sure to adjust the constant term accordingly on the other side of the equation.

The quadratic formula provides a universal solution method: x = $-b ± √(b² - 4ac)$ /2a. This formula works for any quadratic equation in standard form and is especially useful when factoring seems difficult.

Remember: Geometry can help with quadratics too! Similar triangles give us proportional sides, and when two lines intersect in a circle, the products of line segments follow the pattern AP × PB = CP × PD.

Important Tips and Shortcuts

When solving quadratics for real-world measurements like length, always discard negative answers as they rarely make physical sense. Be careful with your algebraic manipulations - never divide an expression by a variable or leave a square root in the denominator.

To rationalize expressions with square roots in the denominator, multiply both numerator and denominator by the denominator itself. If the denominator contains more than one term with a square root $like A + √B$ , multiply by its conjugate $A - √B$ to eliminate the radicals.

When working with negative numbers under a square root, remember to use the imaginary number i. If your solution doesn't check when plugged back into the original equation, try the alternative answer from your solving process.

Study Hack: Memorizing perfect squares up to 20² can save you valuable time on tests! For example, knowing that 15² = 225 or 19² = 361 lets you quickly identify when expressions can be simplified.

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Algebra 2

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131

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Jan 30, 2026

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3 pages

Master Quadratic Equations: Easy Methods Explained

Jenna

@jenna_bmem

Quadratic equations are powerful mathematical tools that appear in countless real-world applications. They're equations containing a squared term as the highest power, and mastering different solving methods will help you tackle many problems in math, physics, and beyond.

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Understanding and Solving Quadratics

Pro Tip: If the discriminant $b² - 4ac$ is a perfect square, the quadratic is likely factorable - making the factoring method your fastest approach!

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Completing the Square and Quadratic Formula

Remember: Geometry can help with quadratics too! Similar triangles give us proportional sides, and when two lines intersect in a circle, the products of line segments follow the pattern AP × PB = CP × PD.

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Access to all documents

Improve your grades

Join milions of students

Important Tips and Shortcuts

Study Hack: Memorizing perfect squares up to 20² can save you valuable time on tests! For example, knowing that 15² = 225 or 19² = 361 lets you quickly identify when expressions can be simplified.