To solve the quadratic equation [tex]\( m^2 - 14m + 49 = 0 \)[/tex], follow these steps:
1. Identify the coefficients:
The given quadratic equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -14 \)[/tex]
- [tex]\( c = 49 \)[/tex]
2. Find the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[
\Delta = (-14)^2 - 4 \cdot 1 \cdot 49
\][/tex]
Calculate [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = 196 - 196 = 0
\][/tex]
3. Determine the nature of the roots:
Since the discriminant [tex]\(\Delta\)[/tex] is 0, the quadratic equation has exactly one real root (a repeated root).
4. Calculate the root:
The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are given by the formula:
[tex]\[
m = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\(\Delta = 0\)[/tex], the formula simplifies to:
[tex]\[
m = \frac{-b}{2a}
\][/tex]
Substitute [tex]\( b = -14 \)[/tex] and [tex]\( a = 1 \)[/tex]:
[tex]\[
m = \frac{-(-14)}{2 \cdot 1} = \frac{14}{2} = 7
\][/tex]
5. Confirm the solution:
Since the discriminant [tex]\(\Delta\)[/tex] is 0, the solution [tex]\( m = 7 \)[/tex] is a repeated root.
Therefore, the solution to the quadratic equation [tex]\( m^2 - 14m + 49 = 0 \)[/tex] is:
[tex]\[
m = 7
\][/tex]
This means that 7 is the only root of the equation.
