Title: Quadratic Equation

A quadratic equation in the variable x is an equation that can be written in the form:

where a, b and c represent real number coefficients.

This form is sometimes called the standard form. The term quadratic is used for any equation where the highest power of the variable x is 2. This means an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a of the variable squared cannot be zero, since otherwise it would be a linear equation.

The quadratic formula for an equation written in standard form is shown in Equation.

The expression b² – 4ac shown under the square root sign is called the discriminant.

If the value of the discriminant is positive then the equation has two roots which are different in value.

If the value of the discriminant is zero then the equation has two roots but they are both the same.

If the value of the discriminant is negative then the equation has no roots that are real numbers. (In this case complex numbers may be used but this is beyond the scope of this discussion).

Example 1:

Solve the following quadratic equation using the quadratic formula.

2x² + 7x – 15 = 0

Solution:

In this case a = 2 b = 7 c = – 15

The value of the discriminant is b² – 4ac = 72 – 4(2)(–15) = 169

Example 2:

Solve the following equation using the quadratic formula.

4x² – 20x + 25 = 0

Solution:

In this case a = 4 b = – 20 c = 25

The value of the discriminant is b² – 4ac = 202 – 4(4)(25) = 0

That is, in this case since the value of the discriminant is zero, the two roots of the equation have the same of 2.5.

Example 3:

Solve the following quadratic equation using the quadratic formula.

5x² + 2x + 3 = 0

Solution:

In this case a = 5 b = 2 c = 3

The value of the discriminant is b² – 4ac = 42 – 4(5)(3) = – 44

Since the value of the discriminant is negative, this equation has no roots that are real numbers.

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