Solve the quadratic equation [tex]x^2-10 x+14=-4[/tex] by completing the square.
A) [tex]x=5 \pm \sqrt{7}[/tex]
B) [tex]x=-5 \pm \sqrt{7}[/tex]
C) [tex]x=7 \pm \sqrt{5}[/tex]
D) [tex]x=-7 \pm \sqrt{5}[/tex]
Answer (2)
Rewrite the equation as x 2 − 10 x = − 18 .
Complete the square by adding ( 2 − 10 ) 2 = 25 to both sides: x 2 − 10 x + 25 = − 18 + 25 , which simplifies to ( x − 5 ) 2 = 7 .
Take the square root of both sides: x − 5 = ± 7 .
Solve for x : x = 5 ± 7 . The solution is x = 5 ± 7 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 10 x + 14 = − 4 and asked to solve for x by completing the square. Completing the square is a useful technique for solving quadratic equations, especially when they are not easily factorable.
Isolating the x Terms First, we want to isolate the terms with x on one side of the equation. To do this, we subtract 14 from both sides:
x 2 − 10 x = − 4 − 14
x 2 − 10 x = − 18
Completing the Square Now, we complete the square. To do this, we take half of the coefficient of the x term (which is -10), square it, and add it to both sides of the equation. Half of -10 is -5, and ( − 5 ) 2 = 25 . So we add 25 to both sides:
x 2 − 10 x + 25 = − 18 + 25
x 2 − 10 x + 25 = 7
Rewriting as a Squared Term Next, we rewrite the left side of the equation as a squared term. Since x 2 − 10 x + 25 is a perfect square trinomial, it can be factored as ( x − 5 ) 2 :
( x − 5 ) 2 = 7
Taking the Square Root Now, we take the square root of both sides of the equation:
( x − 5 ) 2 = ± 7
x − 5 = ± 7
Solving for x Finally, we solve for x by adding 5 to both sides:
x = 5 ± 7
Final Answer Therefore, the solutions to the quadratic equation are x = 5 + 7 and x = 5 − 7 .
Examples
Completing the square is a technique used to solve quadratic equations, which have applications in various fields. For example, engineers use quadratic equations to design bridges and calculate the trajectory of projectiles. Architects use quadratic equations to design parabolic arches and other curved structures. In business, quadratic equations can be used to model profit and cost functions, helping companies optimize their operations and maximize profits. Understanding how to solve quadratic equations is therefore essential in many real-world applications.
To solve the equation x 2 − 10 x + 14 = − 4 by completing the square, we isolate the terms, complete the square to find ( x − 5 ) 2 = 7 , and then solve for x , leading to x = 5 ± 7 . The answer is option A) x = 5 ± 7 .
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