Instead of positive and negative area, I think of colors (green/red). Negative area is created by sides of imaginary length. This seems to be overlooked in discussions, but when completing the square we can have "negative" area. Depending on the values of $a$, $b$ and $c$, the solutions can be positive, negative, zero, or complex. Typically, we need negative lengths to fight the area added by the overhang and make the area collapse to zero. It seems strange to have formulas that begin with a negative sign: Practically, we often memorize the equations we're given, but it doesn't mean you can't try a version that makes sense for you. Who says we can't modify equations to fit our thinking? Ideas like "remove $a$ from the equation" and "use the radius, not diameter" simplifies things, and nicknames like "square, overhang, offset" make the parts memorable. The standard quadratic formula is fine, but I found it hard to memorize. a complicated formula on a complicated equation. Which version of the formula should you use? I'd rather use a simple formula on a simple equation, vs. (Verify with wolfram alpha: roots of 3x^2 + 6x + 24 ) Using the radius formula, we get:Īnd to factor the equation (writing it as a set of multiplications) we do: The entire linear coefficient is 2, so the radius is 1. My thought process: first, divide everything by 3. Completing the square and solving gives us: Where $b$ is now the "radius" (not full diameter) of the overhang. This means our starting equation can be written: Since that's the plan, why not write things in terms of the part we want? Let's make $b$ half the overhang: See also quadratic function, discriminant.Summary: The quadratic formula becomes $x = -b \pm \sqrt$ is the piece we move. "x is equal to negative b, plus or minus the square root, of b squared minus 4ac all over 2a." If you know the tune to "Pop goes the weasel," you can also sing the quadratic equation to its tune to help you remember the quadratic equation. This means that when the discriminant is positive, the quadratic will have two solutions - one where you add the square root of the discriminant, and one where you subtract it.īelow is an example of using the quadratic formula:Īlthough the quadratic equation may at first seem daunting to remember, repeated use can help. The discriminant tells us how many solutions the quadratic has. The part of the formula within the radical is called the discriminant: The quadratic formula mainly involves plugging numbers into the equation, but there are a few things you need to know. In that case, you can use algebra to find the zeros. If the quadratic does not contain the ax 2 term, you cannot use the quadratic formula because the denominator of the quadratic formula will equal 0. If a quadratic is missing either the bx or c term, then set b or c equal to 0. Thus, the quadratic formula can be used to determine the zeros of any parabola, as well as give the axis of symmetry of the parabola. Geometrically, these roots represent the points at which a parabola crosses the x-axis. The ± indicates that the quadratic formula has two solutions. Quadratics are polynomials whose highest power term has a degree of 2.Ī, b and c are constants, where a cannot equal 0. It is the solution to the general quadratic equation. The quadratic formula is a formula used to solve quadratic equations. Home / algebra / solving equations / quadratic formula Quadratic formula
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