Quadratic Solver With Graph: Factor, Formula and Vertex

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Opens on Chapter 18’s example: x² − 5x + 6 = 0.

ax² + bx + c = 0

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How it works

First the number under the root, b² − 4ac: positive means two answers, zero means one, negative means none (the curve never reaches zero). The formula x = (−b ± √(b² − 4ac)) ÷ (2a) then gives them. When a = 1 and two whole numbers multiply to c and add to b, the quadratic factors as (x + p)(x + r) and the answers are −p and −r. The lowest (or highest) point, the vertex, sits halfway between the answers at x = −b ÷ (2a). Each answer is checked by putting it back in.

From the book, Chapter 18. x² − 5x + 6 = 0 gives 2 and 3, with its lowest point at x = 2.5 (value −0.25). 2x² + 3x − 5 = 0 has 49 under the root, giving −2.5 and 1. The 96 sq ft pen is 8 by 12 ft; x² + 6x − 16 = 0 gives −8 and 2. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.