How many real solutions does the following quadratic equation have?

25x2+60x+36=0

(1 point)
solution(s)

1 answer

To determine the number of real solutions of the quadratic equation \(25x^2 + 60x + 36 = 0\), we can use the discriminant method. The discriminant \(\Delta\) for a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by the formula:

\[ \Delta = b^2 - 4ac \]

In this case, the coefficients are:

  • \(a = 25\)
  • \(b = 60\)
  • \(c = 36\)

Now, we can calculate the discriminant:

\[ \Delta = 60^2 - 4 \cdot 25 \cdot 36 \]

Calculating \(60^2\):

\[ 60^2 = 3600 \]

Next, calculating \(4 \cdot 25 \cdot 36\):

\[ 4 \cdot 25 = 100 \]

\[ 100 \cdot 36 = 3600 \]

Now, substituting back into the discriminant:

\[ \Delta = 3600 - 3600 = 0 \]

Since the discriminant \(\Delta\) is equal to 0, the quadratic equation has exactly one real solution (a double root).

Therefore, the answer is:

1 real solution.