Hippocampus Algebra: The discriminant

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To use the quadratic formula to a solve quadratic equation, you need to write the equation in standard form. Solve the following problems. Round your answers to the nearest hundredth. Click “Submit” when finished.

The quadratic equations that you have solved so far have had two solutions that are real numbers. For example, the solutions to the equation <EQUATION> are <EQUATION> and <EQUATION>.

But quadratic equations don’t always have two real-number solutions. For example, consider the equation <EQUATION>. Using the quadratic formula, we find that <EQUATION>. <EQUATION> is not a real number, because there is no real number that can be squared to produce a negative number. This means that the equation <EQUATION> has no solutions.

Now let’s solve the equation <EQUATION>. Using the quadratic formula, we find that <EQUATION>. <EQUATION>. So the solution to the equation is <EQUATION> or <EQUATION>. Therefore the equation <EQUATION> has one solution.

In the quadratic formula, the expression under the radical, called the |b| discriminant |/b| , decides the number of real-number solutions that a quadratic equation has. For example, for the equation <EQUATION> the discriminant is <EQUATION> and the equation has two real-number solutions. For the equation <EQUATION>, the discriminant is <EQUATION> and the equation has one solution. For the equation <EQUATION>, the discriminant is <EQUATION> and the equation has no solutions.

So, the discriminant decides the number of real-number solutions to a quadratic equation. If the discriminant is positive, the quadratic equation has two real-number solutions. If the discriminant is zero, the quadratic equation has one real-number solution. If the discriminant is negative, the quadratic equation has no real-number solutions.

What is the discriminant of <EQUATION>? Click “Submit” when finished.

What value of <EQUATION> results in exactly one solution? Click “Submit” when finished.