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What is the quadratic formula? What is it used for?

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2 Answers

The quadratic formula provides a step by step means to find the roots of a second order polynomial of the form Ax^2+Bx+C=0. It always works the same way and is easy to follow. The quadratic formula calculates 2 roots x1 & x2, often written x1,2 x1=(-B+SQRT(B^2-4AC))/2A. X2=(-B-SQRT(B^2-4AC))/2A

The quadratic formula provides the roots (also called zeroes or x-intercepts) of a quadratic equation.  A quadratic equation is a second-degree equation; its highest term is raised to the second power.  Quadratic equations take the form of a parabola.  They open upward if the coefficient of the first (x^2) term is positive and downward if the coefficient is negative.

Consider the following equation:  f(x) = 2x^2 - 30x + 100

We could factor this equation and get (2x - 20) and (x - 5).  We would then set each factor equal to zero and solve for x.  The solutions would be 10 and 5.

Or we could use the quadratic formula.  The variables a, b, and c designate the coefficients for x^2 (a) and x (b), and the constant (c), so a, b, and c are 2, -30, and 100 respectively.

-b +/- SQRT (b^2 - 4ac)

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               2a

30 +/- SQRT (900 - 800)

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                4

30 +/- 10

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      4

Solutions are 10 and 5.  These are the points at which the parabola crosses the x-axis.  (Coordinates (10,0) and (5,0)).

A quadratic equation may have two, one, or no real roots.  It has two real roots (as in the example above) if the discriminant (the part of the formula under the radical) is positive, one real root if the discriminant is zero (the parabola touches the x-axis but doesn't cross it), and no real roots if the discriminant is negative (the parabola doesn't touch the x-axis).  In this case, the roots are imaginary, but you probably don't need to know this yet.