Find the solutions of the following equations by extracting square roots.
1. 2(x+3)^2 = 18
2. 4a^2 - 147 = a^2
3. 54a^2 - 6 = 24
4. 3c^2 - 5 = 25
5. 2x^2 - 32 = 0
Answer (2)
The solutions for the equations after extracting square roots are: 1) x = 0 , − 6 ; 2) a = 7 , − 7 ; 3) a = 3 5 , − 3 5 ; 4) c = 10 , − 10 ; 5) x = 4 , − 4 .
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To solve these equations by extracting square roots, we need to isolate the squared term first and then take the square root of both sides of the equation.
Equation: 2(x+3)^2 = 18 First, divide both sides by 2: ( x + 3 ) 2 = 9 Take the square root of both sides: x + 3 = ± 3 This gives us two possible equations: x + 3 = 3 x + 3 = − 3 Solve for x : x = 0 x = − 6 Therefore, the solutions are x = 0 and x = − 6 .
Equation: 4a^2 - 147 = a^2 Move a 2 to the left side to get: 3 a 2 = 147 Divide both sides by 3: a 2 = 49 Take the square root of both sides: a = ± 7 Therefore, the solutions are a = 7 and a = − 7 .
Equation: 54a^2 - 6 = 24 First, add 6 to both sides to isolate the squared term: 54 a 2 = 30 Divide both sides by 54: a 2 = 54 30 Simplify the fraction: a 2 = 9 5 Take the square root of both sides: a = ± 9 5 a = ± 3 5 Therefore, the solutions are a = 3 5 and a = − 3 5 .
Equation: 3c^2 - 5 = 25 Add 5 to both sides to get: 3 c 2 = 30 Divide both sides by 3: c 2 = 10 Take the square root of both sides: c = ± 10 Therefore, the solutions are c = 10 and c = − 10 .
Equation: 2x^2 - 32 = 0 Add 32 to both sides and then divide by 2: 2 x 2 = 32 x 2 = 16 Take the square root of both sides: x = ± 4 Therefore, the solutions are x = 4 and x = − 4 .
