Quadratic Equations and the Quadratic Formula

ACT Math · Axiom Academy

LESSON Quadratic Equations and the Quadratic Formula Three ways to solve ax^2+bx+c=0 — and one number, the discriminant, that tells you what kind of answer to expect before you solve anything. 1. Standard Form and Factoring Every quadratic equation can be written in standard form : one side equal to zero, terms ordered by descending power of x . Once it's in that shape, the fastest solving method — when the numbers are friendly — is factoring : rewrite the trinomial as two binomials, then use the fact that a product is zero only when one of its factors is zero. Standard form — a , b , c are just numbers, and a can never be 0 Solve x^2+5x+6=0 . Factor the left side: (x+2)(x+3)=0 . Set each factor to zero: x+2=0 or x+3=0 . So x=-2 or x=-3 . Factoring only works when the factors are "nice." The quadratic formula solves any quadratic, every time — because it's built by completing the square on the general equation ax^2+bx+c=0 itself, so it already encodes every possible case. The quadratic formula — solves every quadratic, no factoring required Divide by a , move the constant over, add to both sides to make the left side a perfect square, then take the square root of both sides. The survives that last step — which is exactly why every quadratic has (up to) two roots. Solve x^2-5x+3=0 . Here a=1 , b=-5 , c=3 , so , giving or . 3. Understanding the Discriminant

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