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Title: Quadratic Equation

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

Quadratic Equation

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.

Quadratic 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

Quadratic Equation example

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

Quadratic Equation solution

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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