Solving Quadratic Equations
Aligned to A-REI.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
Quadratics can be solved by factoring, taking square roots, completing the square, or the quadratic formula. Completing the square turns the equation into a perfect square you can undo with a root.
Worked examples
Example 1
Solve x^2 + 6x + 4 = 0 by completing the square. Move the constant: x^2 + 6x = −4. Add (6/2)^2 = 9 to both sides: x^2 + 6x + 9 = 5, so (x + 3)^2 = 5. Then x + 3 = ±√5, giving x = −3 ± √5.
Example 2
Solve 2x² − 7x + 3 = 0 by factoring. Look for two numbers multiplying to (2)(3) = 6 and adding to −7: those are −6 and −1. Split the middle term: 2x² − 6x − x + 3 = 2x(x − 3) − 1(x − 3) = (2x − 1)(x − 3). So x = 1/2 or x = 3.
Example 3
Common mistake: applying the zero-product rule to a nonzero right side. From (x − 2)(x − 5) = 4 you cannot write x − 2 = 4 — the rule says that if a product is ZERO, some factor is zero, and nothing similar is true of 4. Expand and move everything to one side first: x² − 7x + 10 = 4 becomes x² − 7x + 6 = 0, giving (x − 1)(x − 6) = 0 and x = 1 or 6.
Practice questions
- Solve x^2 − 5x + 6 = 0 by factoring.
- Solve x^2 = 20 by taking square roots.
- Solve 2x^2 + 4x − 3 = 0 with the quadratic formula.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Resolver Ecuaciones Cuadráticas
Las cuadráticas se pueden resolver factorizando, sacando raíces cuadradas, completando el cuadrado o con la fórmula cuadrática. Completar el cuadrado convierte la ecuación en un cuadrado perfecto que puedes deshacer con una raíz.
Ejemplo 1: Resuelve x^2 + 6x + 4 = 0 completando el cuadrado. Mueve la constante: x^2 + 6x = −4. Suma (6/2)^2 = 9 a ambos lados: x^2 + 6x + 9 = 5, así que (x + 3)^2 = 5. Entonces x + 3 = ±√5, dando x = −3 ± √5.
Ejemplo 2: Resuelve 2x² − 7x + 3 = 0 factorizando. Busca dos números que multipliquen (2)(3) = 6 y sumen −7: son −6 y −1. Separa el término central: 2x² − 6x − x + 3 = 2x(x − 3) − 1(x − 3) = (2x − 1)(x − 3). Así que x = 1/2 o x = 3.
Ejemplo 3: Error común: aplicar la regla del producto cero cuando el lado derecho no es cero. De (x − 2)(x − 5) = 4 no puedes escribir x − 2 = 4 — la regla dice que si un producto es CERO, algún factor es cero, y nada parecido vale para el 4. Desarrolla y pasa todo a un lado primero: x² − 7x + 10 = 4 queda x² − 7x + 6 = 0, dando (x − 1)(x − 6) = 0 y x = 1 o 6.
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