Which of the following quadratic equations has only one root?

Question

4^x2-12x+9=0
x²-3x+3=0
2^x2-x-3=0
2x²-5x+3=0
x²-12x+9=0

 

Answer 1

You can write these equations in vertex form.  If the vertex of the equation has a y-coordinate of 0, then the equation has only one root.

 

In vertex form,

 

a(x - h)² + k

 

where:

a is the leading coefficient

(h, k) is the vertex

 

If the value of k is zero, then the equation has one root.

 

 

I will show you the first equation.

 

4x² - 12x + 9 = 4(x² - 3x + (9/4)) = 4(x - 3/2)² + 0

 

Since the vertex is (3/2, 0), this equation has only one root.

 

 

You can also use the discriminant    b² - 4ac.   It the discriminant is zero, then the equation has only one root.

Answer 2

It must be a perfect square.  You can try factoring each equation to see if you have a perfect square, or you can evaluate the discriminant of each one.  If the discriminant is zero, there is one real root to the equation.  Try this approach.

 

f(x) = ax² + bx + c

D = b² - 4ac

If D = 0, then there is exactly one root, and it is real.