Which value must be added to the expression #x^2 - 3x# to make it a perfect-square trinomial?

1 Answer
Jun 9, 2016

Use the discriminant to determine this.

Explanation:

Consider the quadratic equation #x^2 + 6x + 9 = 0#. What would be the solutions to this equation?

Solve by factoring:

#x^2 + 6x + 9 = 0#

#(x + 3)(x + 3) = 0#

#x = -3 and -3#

There is only one solution!

Now, recall that a solution to any equation occurs when #y = 0#. Therefore, for a regular quadratic equation, for example #0 = x^2 + 6x + 5#, there would be two solutions. However, for the example above, there is only one solution. Why?

Because the vertex (a single point, the lowest point on the parabola) lies on the x axis. Therefore, there will only be one solution.

The discriminant is used to calculate the number of solutions to a quadratic equation.

The discriminant, for an equation #0 = ax^2 + bx + c#, is #b^2 - 4ac#. When there are no solutions, the number given by the discriminant will be less than 0. When there are two solutions, the number given by the discriminant will be more than zero. However, if there is but one solution, the number given by the discriminant will be 0. Therefore, we can determine the missing term in your equation by setting the discriminant to 0 and solving for #c#, or the constant term, which is the one we don't know.

Let the constant term be #n#.

Then #a =1, b = -3 and c = n#

#b^2 - 4ac = 0#

#(-3)^2 - (4 xx 1 xx n) = 0#

#9 - 4n = 0#

#-4n = -9#

#n = 9/4#

Therefore, the perfect square trinomial is #x^2 - 3x + 9/4#

Hopefully this helps!