Enter the correct value so that each expression is a perfect-square trinomial.
[tex]\begin{array}{l}
x^2-10 x+\square \\
x^2+\square x+36 \\
x^2+\frac{1}{2} x+\square\end{array}[/tex]
Answer (2)
To make x 2 − 10 x + □ a perfect square trinomial, we find the missing term by ( 2 − 10 ) 2 = 25 .
To make x 2 + □ x + 36 a perfect square trinomial, we find the missing term by 2 ⋅ ( ± 36 ) = ± 12 .
To make x 2 + 2 1 x + □ a perfect square trinomial, we find the missing term by ( 2 1/2 ) 2 = 16 1 .
The missing values are 25 , ± 12 , 16 1 .
Explanation
Understanding the Problem We are given three incomplete quadratic expressions and asked to find the missing term that makes each a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial.
General Form of Perfect Square Trinomial The general form of a perfect square trinomial is x 2 + 2 a x + a 2 or x 2 − 2 a x + a 2 , which factors to ( x + a ) 2 or ( x − a ) 2 , respectively. We will use this form to find the missing terms in each expression.
Finding the Missing Term for the First Expression For the first expression, x 2 − 10 x + □ , we want to find a value such that the expression is a perfect square. Comparing this to x 2 − 2 a x + a 2 , we have − 10 = − 2 a , which means a = 5 . Therefore, the missing term is a 2 = 5 2 = 25 . So the perfect square trinomial is x 2 − 10 x + 25 = ( x − 5 ) 2 .
Finding the Missing Term for the Second Expression For the second expression, x 2 + □ x + 36 , we want to find a value such that the expression is a perfect square. Comparing this to x 2 + 2 a x + a 2 , we have 36 = a 2 , which means a = ± 6 . Therefore, the missing term is 2 a x = 2 ( ± 6 ) x = ± 12 x . So the perfect square trinomials are x 2 + 12 x + 36 = ( x + 6 ) 2 and x 2 − 12 x + 36 = ( x − 6 ) 2 .
Finding the Missing Term for the Third Expression For the third expression, x 2 + 2 1 x + □ , we want to find a value such that the expression is a perfect square. Comparing this to x 2 + 2 a x + a 2 , we have 2 1 = 2 a , which means a = 4 1 . Therefore, the missing term is a 2 = ( 4 1 ) 2 = 16 1 . So the perfect square trinomial is x 2 + 2 1 x + 16 1 = ( x + 4 1 ) 2 .
Final Answer Therefore, the missing values are 25, ± 12 , and 16 1 .
Examples
Perfect square trinomials are useful in various applications, such as completing the square to solve quadratic equations, simplifying algebraic expressions, and solving problems in physics and engineering. For example, in projectile motion, the height of an object can be modeled by a quadratic equation, and completing the square can help determine the maximum height reached by the object. Also, in electrical engineering, perfect square trinomials can be used to analyze circuits and determine the power dissipated in a resistor.
The missing values to make each expression a perfect square trinomial are 25, ±12, and 1/16. For the first expression x 2 − 10 x + 25 , the second can be x 2 ± 12 x + 36 , and the third is x 2 + 2 1 x + 16 1 . These expressions can all be factored into perfect squares.
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