Solve for $x$ in the equation $x^2-10 x+25=0$.
A. $x=5 \pm 2 \sqrt{5}$
B. $x=5 \pm \sqrt{35}$
C. $x=10 \pm 2 \sqrt{5}$
D. $x=10 \pm \sqrt{35}$
Answer (2)
Recognize the quadratic equation as a perfect square trinomial: x 2 − 10 x + 25 = ( x − 5 ) 2 .
Set the perfect square equal to zero: ( x − 5 ) 2 = 0 .
Take the square root of both sides: x − 5 = 0 .
Solve for x : x = 5 , so the final answer is 5 .
Explanation
Problem Analysis We are given the quadratic equation x 2 − 10 x + 25 = 0 . Our goal is to find the value(s) of x that satisfy this equation.
Factoring the Quadratic Notice that the quadratic expression x 2 − 10 x + 25 is a perfect square trinomial. It can be factored as ( x − 5 ) 2 . So, the equation becomes ( x − 5 ) 2 = 0 .
Taking the Square Root To solve for x , we take the square root of both sides of the equation: ( x − 5 ) 2 = 0 , which simplifies to x − 5 = 0 .
Solving for x Finally, we isolate x by adding 5 to both sides of the equation: x = 5 . Therefore, the solution to the equation x 2 − 10 x + 25 = 0 is x = 5 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling growth and decay processes. For instance, if you're designing a bridge, you might use a quadratic equation to model the curve of an arch, ensuring it can withstand specific loads and stresses. Similarly, in business, quadratic equations can help model profit margins or optimize production costs.
The solution to the equation x 2 − 10 x + 25 = 0 is x = 5 . This equation is a perfect square trinomial that simplifies to ( x − 5 ) 2 = 0 leading to the repeated root. None of the provided choices match this answer.
;
