The equation is already in the standard form of , where the discriminant is defined as: 

Following the coefficients of each term with the polynomial in standard form, we can determine the values of each variable.

Substitute the terms into the equation.

The answer is:  

We will need to use the roots to write the quadratic equation.

Simplify by using the FOIL method.

The discriminant of the parabola in standard form is defined as:  

where .

Substitute the values in the equation.

The answer is:  

The given equation is not fully simplified in standard form:   

Distribute the negative three across each term in the parentheses.

Write the formula for the discriminant.

Substitute the known terms.

The answer is:  

The equation is already in standard form .

Following the coefficients: 

The discriminant is the term inside the square root of the quadratic equation.

Substitute the values.

The discriminant is:  

Given the roots, we can set up the binomials of the parabola.

Use the FOIL method to simplify these binomials.

Simplify all the terms by distribution.

The equation of the parabola in standard form is: 

The equation of the discriminant is the square root of the term under the radical of the quadratic equation.  Substitute the terms into the equation.

The answer is:  

Write the formula for the discriminant.  This is the term inside the radical of the quadratic equation.

Substitute the values into the equation.

The answer is:  

The equation is already in standard form:  

Identify the coefficients.

The formula for the discriminant is:

Substitute the values into the equation.

The answer is:  

Write the formula for the discriminant.  The discriminant is the term inside the square root portion of the quadratic formula.

Substitute the known coefficients of the polynomial in standard form:

The answer is:  

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If we were to find the roots of this equation we would first set  and attempt to factor:  

The trivial solution is , the other possible non-zero solutions will depend on the quadratic factor. A quadratic equation may have either 2 complex roots, 2 real roots, or 1 real root. The conditions determined by a the radical term in the quadratic equation...the discriminate: 

For a quadratic equation, or quadratic factor, of the form  the discriminate is defined as: 

For the quadratic factor in this problem we have, 

The discriminate in this case is then,  

Now let's recall how the discriminate determines the number and types of roots 

a) Two Real Roots 

When the discriminate is greater than zero the function has 2 real-valued roots. 

b) One Real Root 

When the discriminate is zero, the function has 1 real valued root: 

Apply to the discriminate and solve for . 

c) Two Complex Roots 

Similarly, for two complex roots solve  , we have: 

 Summary 

a)  

b)  

c) 

                                         (1)

                                             (2)

The first step is conceptualize. What kind of equation is equation (1)? If we rearrange it's clearly a quadratic equation. 

                             (3)

To determine the number of solutions use the discriminate: 

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Reminder

Recall that for a quadratic equation   the general formula for the solution in terms of the constant coefficients is given by:  

The quantity under the radical is known as the discriminate. If the discriminate is less than zero, there are two complex valued solution. In cases where the discriminate is zero, there is just one real solution . If the discriminate is positive, then there are two real solutions. 

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 The discriminate for equation (3) is written: 

We are given that  .  Therefore we have: 

The discriminate is therefore zero, meaning there is only one real solution.