Question

a student solves the equation ( x^2 + 18x = 8x + 3 ). the table shows the student’s work.

step	given	equation
2		( x^2 + 10x = 3 )
3		( x^2 + 10x + 25 = 3 + 25 )
4		( (x + 5)^2 = 28 )
5		( x = -5 - sqrt{28} ) or ( x = -5 + sqrt{28} )

which equivalent equation could complete the table correctly?

• ( x^2 = 23 )
• ( x^2 = 28 )
• ( x + 5 = pm sqrt{23} )
• ( x + 5 = pm sqrt{28} )

Explanation:

Step1: Analyze Step 3 and Step 5

From Step 3, we have \((x + 5)^2=28\). To solve for \(x\), we take the square root of both sides.

Step2: Apply Square Root Property

The square root property states that if \(y^2 = a\), then \(y=\pm\sqrt{a}\). Here, \(y = x + 5\) and \(a = 28\), so applying the property gives \(x + 5=\pm\sqrt{28}\). This equation would be the one to complete Step 4 (the "?") in the table, as it leads to the solutions in Step 5.

Answer:

\(x + 5=\pm\sqrt{28}\) (the fourth option in the given choices)