Answer :
To find the solution(s) to the quadratic equation [tex]\( x^2 - 14x + 49 = 0 \)[/tex], let's follow the standard approach using the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, we have the quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -14 \)[/tex]
- [tex]\( c = 49 \)[/tex]
Let’s begin by calculating the discriminant ([tex]\( \Delta \)[/tex]) of the quadratic equation:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-14)^2 - 4 \cdot 1 \cdot 49 \][/tex]
[tex]\[ \Delta = 196 - 196 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant is 0, which indicates that the quadratic equation has exactly one real solution (a repeated root).
Using the quadratic formula and knowing that [tex]\( \Delta = 0 \)[/tex], we find the solution [tex]\( x \)[/tex] as follows:
[tex]\[ x = \frac{-b \pm \sqrt{0}}{2a} \][/tex]
[tex]\[ x = \frac{-b \pm 0}{2a} \][/tex]
[tex]\[ x = \frac{-(-14)}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{14}{2} \][/tex]
[tex]\[ x = 7 \][/tex]
Therefore, the only solution to the quadratic equation [tex]\( x^2 - 14x + 49 = 0 \)[/tex] is:
[tex]\[ x = 7 \][/tex]
From the given options, the correct answer is:
A. [tex]\( x = 7 \)[/tex] only
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, we have the quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -14 \)[/tex]
- [tex]\( c = 49 \)[/tex]
Let’s begin by calculating the discriminant ([tex]\( \Delta \)[/tex]) of the quadratic equation:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-14)^2 - 4 \cdot 1 \cdot 49 \][/tex]
[tex]\[ \Delta = 196 - 196 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant is 0, which indicates that the quadratic equation has exactly one real solution (a repeated root).
Using the quadratic formula and knowing that [tex]\( \Delta = 0 \)[/tex], we find the solution [tex]\( x \)[/tex] as follows:
[tex]\[ x = \frac{-b \pm \sqrt{0}}{2a} \][/tex]
[tex]\[ x = \frac{-b \pm 0}{2a} \][/tex]
[tex]\[ x = \frac{-(-14)}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{14}{2} \][/tex]
[tex]\[ x = 7 \][/tex]
Therefore, the only solution to the quadratic equation [tex]\( x^2 - 14x + 49 = 0 \)[/tex] is:
[tex]\[ x = 7 \][/tex]
From the given options, the correct answer is:
A. [tex]\( x = 7 \)[/tex] only
