It appears there may be some typographical errors in your question, but I'll address what seems to be a quadratic equation. Let's solve the equation:
[tex]\[ 25x(x + 1) + 4 = 0. \][/tex]
Here are the step-by-step instructions for solving this quadratic equation:
1. Expand and Set the Equation to Standard Form:
The given equation is:
[tex]\[ 25x(x + 1) + 4 = 0. \][/tex]
First, distribute the [tex]\( 25x \)[/tex]:
[tex]\[ 25x^2 + 25x + 4 = 0. \][/tex]
2. Identify Coefficients:
Compare the equation [tex]\( 25x^2 + 25x + 4 = 0 \)[/tex] with the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. From this, we identify:
[tex]\[ a = 25, \][/tex]
[tex]\[ b = 25, \][/tex]
[tex]\[ c = 4. \][/tex]
3. Use the Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ x = \frac{-25 \pm \sqrt{25^2 - 4 \cdot 25 \cdot 4}}{2 \cdot 25}. \][/tex]
4. Calculate the Discriminant:
Compute the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = 25^2 - 4 \cdot 25 \cdot 4 = 625 - 400 = 225. \][/tex]
5. Solve for [tex]\( x \)[/tex]:
Since the discriminant is positive, we will have two real solutions:
[tex]\[ x = \frac{-25 \pm \sqrt{225}}{50}. \][/tex]
[tex]\[ x = \frac{-25 \pm 15}{50}. \][/tex]
This simplifies to two cases:
- Case 1 (Positive Root):
[tex]\[ x = \frac{-25 + 15}{50} = \frac{-10}{50} = -0.2. \][/tex]
- Case 2 (Negative Root):
[tex]\[ x = \frac{-25 - 15}{50} = \frac{-40}{50} = -0.8. \][/tex]
6. Solution:
Therefore, the solutions to the equation [tex]\( 25x(x + 1) + 4 = 0 \)[/tex] are:
[tex]\[ x = -0.2 \quad \text{and} \quad x = -0.8. \][/tex]
These are the values of [tex]\( x \)[/tex] that satisfy the given equation.
