Trinomials are not constantly basic to element. In fact, some trinomials cannot be factored. Thus, we need a different way to settle quadratic equations. Herein lies the prominence of the quadratic formula: Given a quadratic equation ax2 + bx + c = 0, the options are offered by the equation

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 x = Example 1: Solve for x: x2 + 8x + 15.75 = 0a = 1, b = 8, and c = 15.75.x = = = = = or = - or- Thus, x = - or x = - .

Example 2: Solve for x: 3x2 - 10x - 25 = 0.a = 3, b = - 10, and also c = - 25.x = = = = = = or = 5 or- Thus, x = 5 or x = - .

Example 3: Solve for x: -3x2 - 24x - 48 = 0.a = - 3, b = - 24, and c = - 48.x = = = = = = = - 4 Therefore, x = - 4.

Example 4: Solve for x: 2x2 - 4x + 7.a = 2, b = - 4, and also c = 7.x = = = = Because we cannot take the square root of a negative number, tbelow are no options. (The graph of this quadratic polynomial will therefore be a parabola that never before touches the x-axis.)

### The Discriminant

As we have viewed, tbelow have the right to be 0, 1, or 2 services to a quadratic equation, depending on whether the expression inside the square root authorize, (b2 - 4ac), is positive, negative, or zero. This expression has actually a distinct name: the discriminant.

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