Solve the equation: [tex]r^2-5 r+25=0[/tex]. Fully simplify all answers, including non-real solutions.
[tex]r=[/tex]

Apply the quadratic formula r = 2 a − b ± b 2 − 4 a c ​ ​ with a = 1 , b = − 5 , and c = 25 .
Substitute the values into the formula: r = 2 ( 1 ) 5 ± ( − 5 ) 2 − 4 ( 1 ) ( 25 ) ​ ​ .
Simplify the expression: r = 2 5 ± − 75 ​ ​ .
Express the solutions in terms of i : r = 2 5 ​ ± 2 5 3 ​ ​ i .

The solutions are r = 2 5 ​ ± 2 5 3 ​ ​ i ​ .
Explanation

Understanding the Problem We are given the quadratic equation r 2 − 5 r + 25 = 0 . Our goal is to find the values of r that satisfy this equation. Since this is a quadratic equation, we can use the quadratic formula to find the solutions.

Applying the Quadratic Formula The quadratic formula is given by: r = 2 a − b ± b 2 − 4 a c ​ ​ where a , b , and c are the coefficients of the quadratic equation a r 2 + b r + c = 0 . In our case, a = 1 , b = − 5 , and c = 25 .

Substituting the Values Now, we substitute the values of a , b , and c into the quadratic formula: r = 2 ( 1 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 1 ) ( 25 ) ​ ​ r = 2 5 ± 25 − 100 ​ ​ r = 2 5 ± − 75 ​ ​

Simplifying the Square Root Since we have a negative number under the square root, we will have complex solutions. We can rewrite − 75 ​ as 75 ​ i , where i is the imaginary unit ( i 2 = − 1 ). 75 ​ = 25 ⋅ 3 ​ = 5 3 ​ So, − 75 ​ = 5 3 ​ i .

Final Solutions Now we can write the solutions as: r = 2 5 ± 5 3 ​ i ​ r = 2 5 ​ ± 2 5 3 ​ ​ i Thus, the two solutions are r = 2 5 ​ + 2 5 3 ​ ​ i and r = 2 5 ​ − 2 5 3 ​ ​ i .

Conclusion The solutions to the equation r 2 − 5 r + 25 = 0 are: r = 2 5 ​ + 2 5 3 ​ ​ i , r = 2 5 ​ − 2 5 3 ​ ​ i


Examples
Quadratic equations are not just abstract math; they appear in physics, engineering, and even economics. For example, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the arches, ensuring stability and optimal load distribution. Similarly, in finance, quadratic equations can help model investment growth or calculate break-even points. Understanding how to solve them is a fundamental skill that opens doors to solving real-world problems in various fields.

Answered by GinnyAnswer | 2025-07-07

The solutions to the equation r 2 − 5 r + 25 = 0 are r = 2 5 ​ + 2 5 3 ​ ​ i and r = 2 5 ​ − 2 5 3 ​ ​ i , which are complex numbers. We find these solutions using the quadratic formula. Both solutions indicate that the roots of the equation are non-real due to the negative discriminant.
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Answered by Anonymous | 2025-07-13