How many times does the quadratic function below intersect the x-axis?

Question

How many times does the quadratic function below intersect the x-axis?

 

y= x^2 + 10x +25

 

A.) 2

 

B.) 3

 

C.) 1

 

D.) 0

Answer 1

To expand on below, for any quadratic equation:

 

ax^2 + bx + c = 0

 

 

The x-intercepts can be found using the formula:

 

x =  [-b ± √(b² - 4ac)]         = -5 ± 0  = -5

      _________________

                   2a

 

 

With this formula, there are 3 possibilities:

 

1. you get one answer (like here). this means there is only 1 x-intercept

2. you get a ± in your answer, meaning there are 2 x-intercepts

3. you get √(negative #), meaning there are no x-intercepts because the answer is imaginary. 

Answer 2

By factorization,

 

x^2 + 10x + 25 =  0  is equivalent to 

( x + 5)^2   = 0   therefore by finding the square root of both sides we get

x   +   5 =0    therefore,  x  =  -5

 

There is only one root in a quadratic equation  when it is in the form  ax^2 + bx + c = 0  and  b^2 = 4ac

 

 

Similarly,  using the quadratic formula,  x = ( -b ‡ √(b²  - 4ac) ) / 2

 

x = (-10 ‡ √(10² – 4•1•25))  / 2

x  = (-10 ‡ √0 ) /2       =  -10/2   = -5

 

x has only one intersection,  at x = -5

Answer 3

Graphing the equation will show that it will intersect the x-axis only once.

 

You can also use the quadratic equation to find the roots (where they cross the x-axis): x = -10/2 ± ½√(10²-4(1)(25)) = -5 ± 0 = -5.

 

You can also factor: x² + 10x + 25 = (x + 5)² = 0 will yield x = -5.