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Solve for [tex]$x$[/tex] in the equation [tex]$x^2 + 11x + \frac{121}{4} = \frac{125}{4}$[/tex].

A. [tex][tex]$x = -11 \pm \frac{25}{2}$[/tex][/tex]

B. [tex]$x = -\frac{11}{2} \pm \frac{25}{2}$[/tex]

C. [tex]$x = -11 \pm \frac{5 \sqrt{5}}{2}$[/tex]

D. [tex]$x = -\frac{11}{2} \pm \frac{5 \sqrt{5}}{2}$[/tex]

Sagot :

GinnyAnswer
To solve the equation [tex]\( x^2 + 11x + \frac{121}{4} = \frac{125}{4} \)[/tex], let's follow a step-by-step approach to isolate [tex]\( x \)[/tex]:

### Step 1: Simplify and Rearrange the Equation

Given equation:
[tex]\[ x^2 + 11x + \frac{121}{4} = \frac{125}{4} \][/tex]

Subtract [tex]\(\frac{125}{4}\)[/tex] from both sides to set the equation to zero:
[tex]\[ x^2 + 11x + \frac{121}{4} - \frac{125}{4} = 0 \][/tex]
[tex]\[ x^2 + 11x + \frac{121 - 125}{4} = 0 \][/tex]
[tex]\[ x^2 + 11x + \frac{-4}{4} = 0 \][/tex]
[tex]\[ x^2 + 11x - 1 = 0 \][/tex]

This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 11 \)[/tex], and [tex]\( c = -1 \)[/tex].

### Step 2: Solve the Quadratic Equation Using the Quadratic Formula

The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = 11 \)[/tex], and [tex]\( c = -1 \)[/tex] into the formula:
[tex]\[ x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-11 \pm \sqrt{121 + 4}}{2} \][/tex]
[tex]\[ x = \frac{-11 \pm \sqrt{125}}{2} \][/tex]

### Step 3: Express [tex]\(\sqrt{125}\)[/tex] in Simpler Terms

[tex]\[ \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \][/tex]

So, the expression becomes:
[tex]\[ x = \frac{-11 \pm 5\sqrt{5}}{2} \][/tex]

### Step 4: Interpret the Results

We now have two possible solutions:
[tex]\[ x_1 = \frac{-11 + 5\sqrt{5}}{2} \][/tex]
[tex]\[ x_2 = \frac{-11 - 5\sqrt{5}}{2} \][/tex]

These are the solutions to the quadratic equation.

### Final Values

So, the solutions are:
[tex]\[ x = -\frac{11}{2} + \frac{5\sqrt{5}}{2} \][/tex]
[tex]\[ x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \][/tex]

Therefore, the solutions for [tex]\( x \)[/tex] in the equation [tex]\( x^2 + 11x + \frac{121}{4} = \frac{125}{4} \)[/tex] are:
[tex]\[ x = -\frac{11}{2} + \frac{5\sqrt{5}}{2} \][/tex]
[tex]\[ x = -\frac{11}{2} - \frac{5\sqrt{5}}{2} \][/tex]

Alternatively written as:
[tex]\[ x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2} \][/tex]

These solutions match the choices provided in the problem.
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