The Quadratic Formula

Algebra 1 · Axiom Academy

Derive the most powerful tool in algebra by completing the square — one formula that solves every quadratic equation. Every quadratic equation can be written in standard form , where a , b , and c are constants and . Solving it means finding the x -values where the parabola crosses the x -axis. Our goal is to solve for x . First move the constant to the right side, then divide everything by a so the coefficient of x^2 becomes exactly 1 — that's what makes completing the square possible. Here's the key move. To turn the left side into a perfect square , take half the coefficient of x — which is — square it, and add it to both sides. Geometrically you're adding the one missing corner tile that completes a square. With a perfect square on the left, take the square root of both sides. The square root introduces a — this is where the two solutions are born. Then subtract to leave x alone. Combining the two fractions over the common denominator 2a gives the famous Quadratic Formula . This single expression works for any quadratic in standard form — just read off a , b , c and plug in. The expression under the square root, b^2-4ac , is called the discriminant . Its sign alone tells you how many real solutions the formula will produce — because a negative under a square root has no real value. Two different real solutions — the parabola crosses the x -axis twice. One repeated solution — the parabola just touches the x -axis at its vertex.

This is the written version of the interactive lesson above. See the full Algebra 1 course.