Drag the tiles to the correct boxes to complete the pairs. Solve each equation using the square root property. Then, match the solutions to the equations they describe.
Tiles:
$\sqrt{12}-\sqrt{13} \quad \sqrt{14},-\sqrt{14} \quad 91,-91 \quad 21,-21$
Pairs:
$\begin{array}{l}
x^2+81=0 \\
x^2-22=-26 \longrightarrow \\
3 x^2-18=21 \longrightarrow \\
2 x^2+15=-13 \longrightarrow
\end{array}$
Answer (2)
The only equation with real solutions is 3 x 2 − 18 = 21 , which matches with 13 , − 13 . To assume corrections, hypothetical matches could involve real solution equations like x 2 − 441 = 0 matching with 21 , − 21 . Other equations provided do not yield real solutions based on the original context.
;
Solve 3 x 2 − 18 = 21 to get x = ± 13 .
Match 3 x 2 − 18 = 21 with 13 , − 13 .
Note that the other equations do not have real solutions that match the provided tiles.
Assuming typos, match x 2 + 81 = 0 with 21 , − 21 , x 2 − 22 = − 26 with 14 , − 14 , and 2 x 2 + 15 = − 13 with 12 − 13 .
Explanation
Analyze the problem We are given four equations to solve using the square root property and four pairs of solutions to match with the equations. The equations are:
x 2 + 81 = 0
x 2 − 22 = − 26
3 x 2 − 18 = 21
2 x 2 + 15 = − 13
The possible solutions are: 12 , − 12 , 13 , − 13 , 14 , − 14 , 21 , − 21 .
Solve the equations Let's solve each equation:
x 2 + 81 = 0 ⟹ x 2 = − 81 ⟹ x = ± − 81 = ± 9 i . This does not match any of the tiles.
x 2 − 22 = − 26 ⟹ x 2 = − 4 ⟹ x = ± − 4 = ± 2 i . This does not match any of the tiles.
3 x 2 − 18 = 21 ⟹ 3 x 2 = 39 ⟹ x 2 = 13 ⟹ x = ± 13 . This matches the tile 13 , − 13 .
2 x 2 + 15 = − 13 ⟹ 2 x 2 = − 28 ⟹ x 2 = − 14 ⟹ x = ± − 14 = ± i 14 . This does not match any of the tiles.
Match the solutions It seems there may be a typo in the tiles provided. Let's assume the tiles are 12 , − 12 , 13 , − 13 , 14 , − 14 , and 21 , − 21 . Also, there might be a typo in the equations. Let's consider a modified set of equations:
x 2 − 441 = 0 ⟹ x 2 = 441 ⟹ x = ± 21 . This matches the tile 21 , − 21 .
3 x 2 − 18 = 21 ⟹ 3 x 2 = 39 ⟹ x 2 = 13 ⟹ x = ± 13 . This matches the tile 13 , − 13 .
Let's assume the second equation is x 2 − 14 = 0 ⟹ x 2 = 14 ⟹ x = ± 14 . This matches the tile 14 , − 14 .
Let's assume we have an equation x 2 − 12 = 0 ⟹ x 2 = 12 ⟹ x = ± 12 . This matches the tile 12 , − 12 .
Final matches Based on the original equations and the given tiles, we have:
3 x 2 − 18 = 21 ⟹ x = ± 13 . So, 3 x 2 − 18 = 21 ⟶ 13 , − 13 .
However, the other equations do not have real solutions that match the provided tiles. Assuming there are typos in the problem, we can make the following matches:
x 2 − 441 = 0 ⟹ x = ± 21 . So, x 2 + 81 = 0 ⟶ 21 , − 21 (assuming the equation was meant to be x 2 − 441 = 0 )
x 2 − 14 = 0 ⟹ x = ± 14 . So, x 2 − 22 = − 26 ⟶ 14 , − 14 (assuming the equation was meant to be x 2 − 14 = 0 )
x 2 − 12 = 0 ⟹ x = ± 12 . So, 2 x 2 + 15 = − 13 ⟶ 12 − 13 (assuming the equation was meant to be x 2 − 12 = 0 )
Examples
The square root property is used in many areas of physics and engineering. For example, when calculating the period of a pendulum, you need to find the square root of the length of the pendulum. Similarly, in electrical engineering, when calculating the impedance of a circuit, you often need to find the square root of the resistance and reactance. Understanding how to solve equations using the square root property is therefore essential for solving real-world problems in these fields. For instance, if you're designing a bridge and need to calculate the tension in a cable, you might use the square root property to solve for the unknown variable. This ensures the bridge is safe and stable.
