Radical Equations

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Answer Key & Explanations

Radical Equations · Grade 9 · Worksheet 2

1. Olivia is examining a square drawn on a coordinate plane with vertices at (0,0), (s,0), (s,s), and (0,s). A circle is inscribed inside this square. The area of the circle is given by the expression 49π square units. What is the value of s? Answer: 14 Solution: The area of a circle is A = πr^2. We are told the area is 49π. So, πr^2 = 49π.
   Full step-by-step solution

   Step 1: The area of a circle is A = πr^2. We are told the area is 49π. So, πr^2 = 49π. Divide both sides by π to get r^2 = 49. Take the square root of both sides: r = 7 (since radius is positive). Step 2: For a circle inscribed in a square, the circle touches all four sides. The diameter of the circle equals the side length of the square. So, s = 2r. Step 3: Substitute r = 7: s = 2 × 7 = 14. The answer is 14.

2. Sophia is a safety inspector testing the braking distance of a new electric vehicle. The braking distance d (in meters) on a dry road is given by the formula d = √(0.5v + 13), where v is the speed of the car in kilometers per hour (km/h) just before braking. During a test, the vehicle's braking distance is measured as exactly 7 meters. What was the speed of the car just before braking, in km/h? Answer: 72 Solution: Write the equation using the given braking distance: √(0.5v + 13) = 7. Square both sides to eliminate the square root: (√(0.5v + 13))² = 7² → 0.5v + 13 = 49. Subtract 13 from both sides: 0.5v = 49 - 13 → 0.5v = 36.
   Full step-by-step solution

   Step 1: Write the equation using the given braking distance: √(0.5v + 13) = 7. Step 2: Square both sides to eliminate the square root: (√(0.5v + 13))² = 7² → 0.5v + 13 = 49. Step 3: Subtract 13 from both sides: 0.5v = 49 - 13 → 0.5v = 36. Step 4: Divide both sides by 0.5: v = 36 / 0.5 → v = 72. Step 5: Check by substituting v = 72 into the original equation: √(0.5(72) + 13) = √(36 + 13) = √49 = 7. The solution is valid. The speed of the car just before braking was 72 km/h.

3. √(5x + 25) - 10 = 0 Answer: 15 Solution: Add 10 to both sides to isolate the radical: √(5x + 25) = 10 Square both sides: (√(5x + 25))² = 10² Simplify: 5x + 25 = 100 Subtract 25 from both sides: 5x = 75 Divide both sides by 5: x = 15 Check the solution in the original equation: √(5(15) + 25) - 10 = √(75 + 25) - 10 = √100 - 10 = 10 - 10…
   Full step-by-step solution

   Step 1: Add 10 to both sides to isolate the radical: √(5x + 25) = 10 Step 2: Square both sides: (√(5x + 25))² = 10² Step 3: Simplify: 5x + 25 = 100 Step 4: Subtract 25 from both sides: 5x = 75 Step 5: Divide both sides by 5: x = 15 Step 6: Check the solution in the original equation: √(5(15) + 25) - 10 = √(75 + 25) - 10 = √100 - 10 = 10 - 10 = 0 The solution checks out. The answer is 15.

4. Mere is examining a square drawn on a coordinate plane with vertices at (0,0), (4,0), (4,4), and (0,4). A circle is inscribed inside the square, touching all four sides. A right triangle is drawn inside the circle such that its hypotenuse is a diameter of the circle and its legs are parallel to the axes. One leg of the triangle has length 2 units. The length of the other leg is described by the radical equation sqrt(4x - 8) = 4. Solve this equation to find the length of the other leg. Answer: 6 Solution: Isolate the radical. The equation is sqrt(4x - 8) = 4. The radical is already isolated.
   Full step-by-step solution

   Step 1: Isolate the radical. The equation is sqrt(4x - 8) = 4. The radical is already isolated. Step 2: Square both sides to eliminate the square root: (sqrt(4x - 8))^2 = 4^2, which gives 4x - 8 = 16. Step 3: Solve for x. Add 8 to both sides: 4x = 24. Divide both sides by 4: x = 6. Step 4: Check the solution in the original equation. Substitute x = 6 into sqrt(4x - 8): sqrt(4*6 - 8) = sqrt(24 - 8) = sqrt(16) = 4. The right side is also 4, so x = 6 is valid. The length of the other leg is 6 units. The answer is 6.

5. A construction company is designing a curved ramp for a skate park. The height of the ramp above the ground is modeled by the function h(x) = √(2x + 11), where x is the horizontal distance in meters from the starting point. If the ramp needs to be exactly 5 meters high at a certain point for a specific trick, what horizontal distance from the start does this occur at? Answer: 7 Solution: Set up the equation using the given height: √(2x + 11) = 5 Square both sides to eliminate the square root: (√(2x + 11))² = 5² Simplify: 2x + 11 = 25 Subtract 11 from both sides: 2x = 14 Divide both sides by 2: x = 7 Check the solution by plugging x = 7 back into the original equation: √(2(7) +…
   Full step-by-step solution

   Step 1: Set up the equation using the given height: √(2x + 11) = 5 Step 2: Square both sides to eliminate the square root: (√(2x + 11))² = 5² Step 3: Simplify: 2x + 11 = 25 Step 4: Subtract 11 from both sides: 2x = 14 Step 5: Divide both sides by 2: x = 7 Step 6: Check the solution by plugging x = 7 back into the original equation: √(2(7) + 11) = √(14 + 11) = √25 = 5 The answer is 7 meters.

6. Emma is a marine biologist tracking the migration of a rare species of sea turtle. The distance d (in kilometers) that a tagged turtle has traveled from its nesting beach is modeled by the equation d = √(3t + 49), where t is the number of days since the turtle was tagged. If the turtle's distance from the beach is exactly 11 kilometers, how many days has it been since the turtle was tagged? Answer: 24 Solution: Write the equation with the given distance: √(3t + 49) = 11 Square both sides to eliminate the square root: (√(3t + 49))² = 11² Simplify: 3t + 49 = 121 Subtract 49 from both sides: 3t = 72 Divide both sides by 3: t = 24 Check by substituting t = 24 back into the original equation: √(3(24) + 49)…
   Full step-by-step solution

   Step 1: Write the equation with the given distance: √(3t + 49) = 11 Step 2: Square both sides to eliminate the square root: (√(3t + 49))² = 11² Step 3: Simplify: 3t + 49 = 121 Step 4: Subtract 49 from both sides: 3t = 72 Step 5: Divide both sides by 3: t = 24 Step 6: Check by substituting t = 24 back into the original equation: √(3(24) + 49) = √(72 + 49) = √121 = 11. The solution is valid. The turtle has been tagged for 24 days.

7. Emma draws a right triangle on a coordinate plane with vertices at (0,0), (b,0), and (0,h). The hypotenuse has length 10 units, and the height h is 2 units longer than the base b. A radical equation that models this situation is sqrt(b^2 + (b + 2)^2) = 10. Solve for b. Answer: 6 Solution: Write the equation: sqrt(b^2 + (b + 2)^2) = 10. Square both sides: b^2 + (b + 2)^2 = 100. Expand (b + 2)^2: (b + 2)^2 = b^2 + 4b + 4.
   Full step-by-step solution

   Step 1: Write the equation: sqrt(b^2 + (b + 2)^2) = 10. Step 2: Square both sides: b^2 + (b + 2)^2 = 100. Step 3: Expand (b + 2)^2: (b + 2)^2 = b^2 + 4b + 4. Step 4: Substitute and combine: b^2 + b^2 + 4b + 4 = 100 -> 2b^2 + 4b + 4 = 100. Step 5: Subtract 100 from both sides: 2b^2 + 4b + 4 - 100 = 0 -> 2b^2 + 4b - 96 = 0. Step 6: Divide the entire equation by 2: b^2 + 2b - 48 = 0. Step 7: Factor the quadratic: (b + 8)(b - 6) = 0. Step 8: Set each factor to zero: b + 8 = 0 -> b = -8; b - 6 = 0 -> b = 6. Step 9: Check both solutions in the original equation. For b = -8: sqrt((-8)^2 + (-8 + 2)^2) = sqrt(64 + (-6)^2) = sqrt(64 + 36) = sqrt(100) = 10, but a length cannot be negative, so b = -8 is extraneous. For b = 6: sqrt(6^2 + (6 + 2)^2) = sqrt(36 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10, which matches. So the only valid solution is b = 6. The answer is 6.