Answer :
To solve the equation [tex]\(x^2 - 26x + 169 = -13\)[/tex] using square roots, let's go through the solution step by step.
1. Rearrange the equation:
Start with the original equation:
[tex]\[
x^2 - 26x + 169 = -13
\][/tex]
Move [tex]\(-13\)[/tex] to the left side to set the equation to zero:
[tex]\[
x^2 - 26x + 169 + 13 = 0
\][/tex]
Simplify this:
[tex]\[
x^2 - 26x + 182 = 0
\][/tex]
2. Solve using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Where [tex]\(a = 1\)[/tex], [tex]\(b = -26\)[/tex], and [tex]\(c = 182\)[/tex].
3. Calculate the discriminant:
Calculate [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
(-26)^2 - 4 \times 1 \times 182 = 676 - 728 = -52
\][/tex]
4. Analyze the discriminant:
Since the discriminant [tex]\(-52\)[/tex] is less than zero, this indicates that there are no real solutions to the given equation.
5. Conclusion:
The equation [tex]\(x^2 - 26x + 182 = 0\)[/tex] has no real solutions because the discriminant is negative. This means the solutions are complex numbers, which are not covered under real number solutions. Therefore, we conclude that there are no real solutions for this equation.
1. Rearrange the equation:
Start with the original equation:
[tex]\[
x^2 - 26x + 169 = -13
\][/tex]
Move [tex]\(-13\)[/tex] to the left side to set the equation to zero:
[tex]\[
x^2 - 26x + 169 + 13 = 0
\][/tex]
Simplify this:
[tex]\[
x^2 - 26x + 182 = 0
\][/tex]
2. Solve using the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Where [tex]\(a = 1\)[/tex], [tex]\(b = -26\)[/tex], and [tex]\(c = 182\)[/tex].
3. Calculate the discriminant:
Calculate [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
(-26)^2 - 4 \times 1 \times 182 = 676 - 728 = -52
\][/tex]
4. Analyze the discriminant:
Since the discriminant [tex]\(-52\)[/tex] is less than zero, this indicates that there are no real solutions to the given equation.
5. Conclusion:
The equation [tex]\(x^2 - 26x + 182 = 0\)[/tex] has no real solutions because the discriminant is negative. This means the solutions are complex numbers, which are not covered under real number solutions. Therefore, we conclude that there are no real solutions for this equation.
