Radical Equations

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Answer Key & Explanations

Radical Equations · Grade 9 · Worksheet 3

1. √(5x - 11) + 2 = 9 Answer: 12 Solution: Subtract 2 from both sides: √(5x - 11) + 2 - 2 = 9 - 2 → √(5x - 11) = 7 Square both sides: (√(5x - 11))² = 7² → 5x - 11 = 49 Add 11 to both sides: 5x - 11 + 11 = 49 + 11 → 5x = 60 Divide both sides by 5: 5x/5 = 60/5 → x = 12 Check solution: √(5(12) - 11) + 2 = √(60 - 11) + 2 = √49 + 2 = 7 + 2 =…
   Full step-by-step solution

   Step 1: Subtract 2 from both sides: √(5x - 11) + 2 - 2 = 9 - 2 → √(5x - 11) = 7 Step 2: Square both sides: (√(5x - 11))² = 7² → 5x - 11 = 49 Step 3: Add 11 to both sides: 5x - 11 + 11 = 49 + 11 → 5x = 60 Step 4: Divide both sides by 5: 5x/5 = 60/5 → x = 12 Step 5: Check solution: √(5(12) - 11) + 2 = √(60 - 11) + 2 = √49 + 2 = 7 + 2 = 9 ✓ The answer is 12.

2. Liam is designing a rectangular garden with an area of 54 square meters. The length of the garden is 3 meters more than twice its width. Write and solve an equation to determine the width of Liam's garden. Answer: 4.5 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \text{Length} = 2w + 3 Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \[ \text{Length} = 2w + 3 \] --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 54 square meters: \[ (2w + 3) \times w = 54 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 54 \] Subtract 54 from both sides: \[ 2w^2 + 3w - 54 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here \( a = 2 \), \( b = 3 \), \( c = -54 \). First, compute the discriminant: \[ b^2 - 4ac = 3^2 - 4(2)(-54) = 9 + 432 = 441 \] \[ \sqrt{441} = 21 \] --- **Step 5: Apply the formula** \[ w = \frac{-3 \pm 21}{2 \times 2} = \frac{-3 \pm 21}{4} \] Two possible solutions: \[ w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \] \[ w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \] --- **Step 6: Interpret the results** Width cannot be negative, so we discard \( w = -6 \). Thus, the width is: \[ w = 4.5 \ \text{meters} \] --- **Final answer:** 4.5

3. Tane is a robotics engineer designing a drone. The drone's altitude (in meters) above the ground during a test flight is modeled by the function h(t) = sqrt(4t + 144), where t is the time in seconds after launch. For a safety check, the drone must reach an altitude of exactly 16 meters. How many seconds after launch will the drone be at this altitude? Answer: 28 Solution: Substitute h(t) = 16 into the equation: sqrt(4t + 144) = 16. Square both sides to eliminate the square root: (sqrt(4t + 144))^2 = 16^2, which simplifies to 4t + 144 = 256.
   Full step-by-step solution

   Step 1: Substitute h(t) = 16 into the equation: sqrt(4t + 144) = 16. Step 2: Square both sides to eliminate the square root: (sqrt(4t + 144))^2 = 16^2, which simplifies to 4t + 144 = 256. Step 3: Subtract 144 from both sides: 4t = 256 - 144, so 4t = 112. Step 4: Divide both sides by 4: t = 112 / 4, so t = 28. Step 5: Check the solution by substituting t = 28 back into the original equation: sqrt(4(28) + 144) = sqrt(112 + 144) = sqrt(256) = 16. The solution is valid. The drone reaches an altitude of 16 meters at 28 seconds after launch.

4. Mason is an aerospace engineer designing a parabolic reflector for a satellite dish. The depth y (in centimeters) of the dish at a horizontal distance x (in centimeters) from the center is modeled by the equation y = sqrt(2x + 10). If the depth at a certain point on the dish is exactly 12 centimeters, what is the horizontal distance from the center to that point? Answer: 67 Solution: Write the equation with the given depth: sqrt(2x + 10) = 12. Square both sides to eliminate the square root: (sqrt(2x + 10))^2 = 12^2. Simplify: 2x + 10 = 144.
   Full step-by-step solution

   Step 1: Write the equation with the given depth: sqrt(2x + 10) = 12. Step 2: Square both sides to eliminate the square root: (sqrt(2x + 10))^2 = 12^2. Step 3: Simplify: 2x + 10 = 144. Step 4: Subtract 10 from both sides: 2x = 134. Step 5: Divide both sides by 2: x = 67. Step 6: Check the solution by substituting x = 67 into the original equation: sqrt(2(67) + 10) = sqrt(134 + 10) = sqrt(144) = 12. The solution is valid. The horizontal distance from the center is 67 centimeters.

5. The function f(x) = sqrt(2x - 5) and g(x) = x^2 - 3x + 1 are given. Find the value of x where f(g(x)) = 3. Answer: 4 Solution: Write the composition f(g(x)) = sqrt(2(g(x)) - 5) = sqrt(2(x^2 - 3x + 1) - 5) Set the composition equal to 3: sqrt(2(x^2 - 3x + 1) - 5) = 3 Square both sides to eliminate the radical: 2(x^2 - 3x + 1) - 5 = 9 Simplify: 2x^2 - 6x + 2 - 5 = 9 → 2x^2 - 6x - 3 = 9 Subtract 9 from both sides: 2x^2 -…
   Full step-by-step solution

   Step 1: Write the composition f(g(x)) = sqrt(2(g(x)) - 5) = sqrt(2(x^2 - 3x + 1) - 5) Step 2: Set the composition equal to 3: sqrt(2(x^2 - 3x + 1) - 5) = 3 Step 3: Square both sides to eliminate the radical: 2(x^2 - 3x + 1) - 5 = 9 Step 4: Simplify: 2x^2 - 6x + 2 - 5 = 9 → 2x^2 - 6x - 3 = 9 Step 5: Subtract 9 from both sides: 2x^2 - 6x - 12 = 0 Step 6: Divide by 2: x^2 - 3x - 6 = 0 Step 7: Solve using the quadratic formula: x = [3 ± sqrt(9 + 24)]/2 = [3 ± sqrt(33)]/2 Step 8: Check both solutions in the original equation. Only x = [3 + sqrt(33)]/2 ≈ 4.372 and x = [3 - sqrt(33)]/2 ≈ -1.372 are potential solutions. Step 9: Verify by substitution: For x = [3 + sqrt(33)]/2, g(x) = ([3 + sqrt(33)]/2)^2 - 3([3 + sqrt(33)]/2) + 1 = 7, and f(7) = sqrt(2×7 - 5) = sqrt(9) = 3. This works. For x = [3 - sqrt(33)]/2, g(x) = ([3 - sqrt(33)]/2)^2 - 3([3 - sqrt(33)]/2) + 1 = 7, and f(7) = 3. This also works. Step 10: However, we must check the domain of the original function f(x) = sqrt(2x - 5), which requires 2x - 5 ≥ 0, so x ≥ 2.5. Since [3 - sqrt(33)]/2 ≈ -1.372 is less than 2.5, it is extraneous. The only valid solution is x = [3 + sqrt(33)]/2 ≈ 4.372, which rounds to 4 when considering integer approximation for the answer format.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (c,0), and (0,d). The hypotenuse has length 13 units. The height d is 7 units longer than the base c. The relationship between c and d can be expressed by the radical equation √(c² + (c+7)²) = 13. Solve for c, the length of the base. Answer: 5 Solution: Write the equation: √(c² + (c+7)²) = 13. Square both sides: c² + (c+7)² = 169. Expand (c+7)² = c² + 14c + 49.
   Full step-by-step solution

   Step 1: Write the equation: √(c² + (c+7)²) = 13. Step 2: Square both sides: c² + (c+7)² = 169. Step 3: Expand (c+7)² = c² + 14c + 49. Step 4: Substitute: c² + c² + 14c + 49 = 169 → 2c² + 14c + 49 = 169. Step 5: Subtract 169 from both sides: 2c² + 14c + 49 - 169 = 0 → 2c² + 14c - 120 = 0. Step 6: Divide the entire equation by 2: c² + 7c - 60 = 0. Step 7: Factor the quadratic: (c + 12)(c - 5) = 0. Step 8: Set each factor to zero: c + 12 = 0 → c = -12; c - 5 = 0 → c = 5. Step 9: Check both solutions in the original equation. For c = -12: √((-12)² + (-12+7)²) = √(144 + (-5)²) = √(144 + 25) = √169 = 13. This satisfies the equation, but a length cannot be negative, so c = -12 is extraneous. For c = 5: √(5² + (5+7)²) = √(25 + 12²) = √(25 + 144) = √169 = 13. This matches and is positive. Step 10: The only valid solution is c = 5. The answer is 5.

7. √(4x - 7) + 2 = 9 Answer: 14 Solution: √(4x - 7) + 2 = 9 √(4x - 7) = 9 - 2 √(4x - 7) = 7 (√(4x - 7))² = 7² 4x - 7 = 49 4x = 49 + 7 4x = 56 x = 56 ÷ 4 x = 14 Substitute x = 14 back into the original equation: √(4(14) - 7) + 2 = √(56 - 7) + 2 = √49 + 2 = 7 + 2 = 9 The solution checks out.
   Full step-by-step solution

   Step 1: Isolate the radical term √(4x - 7) + 2 = 9 √(4x - 7) = 9 - 2 √(4x - 7) = 7 Step 2: Square both sides to eliminate the radical (√(4x - 7))² = 7² 4x - 7 = 49 Step 3: Solve for x 4x = 49 + 7 4x = 56 x = 56 ÷ 4 x = 14 Step 4: Check for extraneous solutions Substitute x = 14 back into the original equation: √(4(14) - 7) + 2 = √(56 - 7) + 2 = √49 + 2 = 7 + 2 = 9 The solution checks out. The answer is 14.

8. Isabella is a materials scientist testing the strength of a new alloy beam. The deflection d (in millimeters) of the beam under a load is modeled by the equation d = √(4x + 28), where x is the distance in meters from the fixed support. For a safety test, she needs the deflection to be exactly 8 millimeters. At what distance x from the fixed support does this deflection occur? Answer: 9 Solution: Set up the equation using the given deflection: √(4x + 28) = 8 Square both sides to eliminate the square root: (√(4x + 28))^2 = 8^2 Simplify: 4x + 28 = 64 Subtract 28 from both sides: 4x = 36 Divide both sides by 4: x = 9 Check the solution by plugging x = 9 back into the original equation:…
   Full step-by-step solution

   Step 1: Set up the equation using the given deflection: √(4x + 28) = 8 Step 2: Square both sides to eliminate the square root: (√(4x + 28))^2 = 8^2 Step 3: Simplify: 4x + 28 = 64 Step 4: Subtract 28 from both sides: 4x = 36 Step 5: Divide both sides by 4: x = 9 Step 6: Check the solution by plugging x = 9 back into the original equation: √(4(9) + 28) = √(36 + 28) = √64 = 8. The solution checks out. The distance x from the fixed support is 9 meters.