Radical Equations

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Radical Equations · Grade 9 · Worksheet 1

1. Mere is a park designer creating a new walking trail. The trail includes a suspension bridge over a small ravine. The height h (in meters) of the main cable above the bridge deck at a horizontal distance x (in meters) from the left tower is modeled by the function h(x) = sqrt(4x + 21). For safety reasons, a support beam must be placed exactly where the cable is 9 meters above the deck. What horizontal distance x from the left tower should Mere place the support beam? Answer: 15 Solution: Set up the equation using the given height. We know h(x) = sqrt(4x + 21) and the height is 9 meters, so: sqrt(4x + 21) = 9.
   Full step-by-step solution

   Step 1: Set up the equation using the given height. We know h(x) = sqrt(4x + 21) and the height is 9 meters, so: sqrt(4x + 21) = 9. Step 2: Square both sides to eliminate the square root: (sqrt(4x + 21))^2 = 9^2, which simplifies to 4x + 21 = 81. Step 3: Subtract 21 from both sides: 4x = 81 - 21, so 4x = 60. Step 4: Divide both sides by 4: x = 60 / 4, so x = 15. Step 5: Check the solution by substituting x = 15 back into the original equation: sqrt(4(15) + 21) = sqrt(60 + 21) = sqrt(81) = 9. The solution is valid. The horizontal distance from the left tower is 15 meters.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle such that it touches all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: A = (0,0) B = (6,0) C = (6,8) AB along the x-axis from (0,0) to (6,0) → length = 6 BC vertical from (6,0) to (6,8) → length = 8 AC is the hypotenuse from (0,0) to (6,8) → length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 a = BC = 8 (opposite A) b = AC = 10 (opposite B) c = AB = 6…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with: AB along the x-axis from (0,0) to (6,0) → length = 6 BC vertical from (6,0) to (6,8) → length = 8 AC is the hypotenuse from (0,0) to (6,8) → length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 So sides: a = BC = 8 (opposite A) b = AC = 10 (opposite B) c = AB = 6 (opposite C) But standard notation: Right angle at B = (6,0) So: AB = 6 (horizontal leg) BC = 8 (vertical leg) AC = 10 (hypotenuse) --- **Step 2: Inradius formula for a right triangle** For a right triangle with legs a and b, hypotenuse c, the inradius r is: r = (a + b - c) / 2 Here: a = 6, b = 8, c = 10 --- **Step 3: Apply formula** r = (6 + 8 - 10) / 2 r = (14 - 10) / 2 r = 4 / 2 r = 2 --- **Step 4: Verify reasoning** The inradius formula for any triangle is r = A / s, where A is area, s is semiperimeter. Area = (1/2) * 6 * 8 = 24 Perimeter = 6 + 8 + 10 = 24 Semiperimeter s = 24 / 2 = 12 So r = A / s = 24 / 12 = 2 Both methods give r = 2. --- **Final Answer:** 2

3. √(2x + 3) = 5 Answer: 11 Solution: sqrt(2x + 3) = 5 Recognize that the square root is already isolated on the left side. To remove the square root, square both sides of the equation. (sqrt(2x + 3))^2 = 5^2 Simplify both sides.
   Full step-by-step solution

   We are solving the equation: sqrt(2x + 3) = 5 Step 1: Recognize that the square root is already isolated on the left side. Step 2: To remove the square root, square both sides of the equation. This gives: (sqrt(2x + 3))^2 = 5^2 Step 3: Simplify both sides. On the left: squaring the square root leaves just the expression inside: 2x + 3. On the right: 5 squared is 25. So we have: 2x + 3 = 25 Step 4: Subtract 3 from both sides to isolate the term with x. 2x + 3 - 3 = 25 - 3 2x = 22 Step 5: Divide both sides by 2 to solve for x. 2x / 2 = 22 / 2 x = 11 Step 6: Check the solution in the original equation to ensure it is valid. Substitute x = 11 into sqrt(2x + 3): sqrt(2*11 + 3) = sqrt(22 + 3) = sqrt(25) = 5 This matches the right-hand side of the original equation, so the solution is correct. Final answer: x = 11

4. √(5x + 25) - 5 = 0 Answer: 0 Solution: Add 5 to both sides to isolate the radical: √(5x + 25) = 5 Square both sides: (√(5x + 25))² = 5² Simplify: 5x + 25 = 25 Subtract 25 from both sides: 5x = 0 Divide both sides by 5: x = 0 Check the solution in the original equation: √(5(0) + 25) - 5 = √(0 + 25) - 5 = √25 - 5 = 5 - 5 = 0 The…
   Full step-by-step solution

   Step 1: Add 5 to both sides to isolate the radical: √(5x + 25) = 5 Step 2: Square both sides: (√(5x + 25))² = 5² Step 3: Simplify: 5x + 25 = 25 Step 4: Subtract 25 from both sides: 5x = 0 Step 5: Divide both sides by 5: x = 0 Step 6: Check the solution in the original equation: √(5(0) + 25) - 5 = √(0 + 25) - 5 = √25 - 5 = 5 - 5 = 0 The solution checks out. The answer is 0.

5. Matiu is designing a rectangular solar panel array for a school project. The area of the array is 80 square meters. The length of the array is 4 meters more than twice its width. To determine the width, Matiu sets up the equation for the length L in terms of width w: L = 2w + 4, and the area equation: w(2w + 4) = 80. Solve for the width w (in meters) of the array. Answer: w = 6 Solution: Write the area equation: w(2w + 4) = 80 Expand: 2w^2 + 4w = 80 Subtract 80 from both sides: 2w^2 + 4w - 80 = 0 Divide the entire equation by 2 to simplify: w^2 + 2w - 40 = 0 Factor the quadratic: (w + 8)(w - 6) = 0 Set each factor equal to zero: w + 8 = 0 gives w = -8; w - 6 = 0 gives w = 6…
   Full step-by-step solution

   Step 1: Write the area equation: w(2w + 4) = 80 Step 2: Expand: 2w^2 + 4w = 80 Step 3: Subtract 80 from both sides: 2w^2 + 4w - 80 = 0 Step 4: Divide the entire equation by 2 to simplify: w^2 + 2w - 40 = 0 Step 5: Factor the quadratic: (w + 8)(w - 6) = 0 Step 6: Set each factor equal to zero: w + 8 = 0 gives w = -8; w - 6 = 0 gives w = 6 Step 7: Since width cannot be negative, the only valid solution is w = 6 Step 8: Check: If w = 6, then length L = 2(6) + 4 = 16, and area = 6 * 16 = 80, which matches. The width of the array is 6 meters.

6. Charlotte is designing a rectangular solar panel. The area of the panel is 72 square meters. The length is 7 meters more than the width. The safety inspector requires that the diagonal of the panel be exactly 17 meters. Write and solve a radical equation to determine the width of the panel. What is the width in meters? Answer: 8 Solution: Let w be the width in meters. The length is w + 7 meters. Using the Pythagorean theorem: w^2 + (w+7)^2 = 17^2.
   Full step-by-step solution

   Let w be the width in meters. The length is w + 7 meters. Using the Pythagorean theorem: w^2 + (w+7)^2 = 17^2. Expand: w^2 + w^2 + 14w + 49 = 289. Combine like terms: 2w^2 + 14w + 49 = 289. Subtract 289 from both sides: 2w^2 + 14w - 240 = 0. Divide by 2: w^2 + 7w - 120 = 0. Factor: (w + 15)(w - 8) = 0. So w = -15 or w = 8. Width cannot be negative, so w = 8. Check: sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17. The width is 8 meters.

7. Noah is analyzing a square drawn on a coordinate plane with vertices at (0,0), (12,0), (12,12), and (0,12). A circle is inscribed in the square, and a smaller square is inscribed inside the circle such that its vertices lie on the circle. If the diagonal of the smaller square is equal to the side length of the larger square, what is the radius of the circle? Answer: 6 Solution: The larger square has side length 12, so its vertices are at (0,0), (12,0), (12,12), and (0,12). The circle is inscribed in the larger square, meaning the circle touches all four sides of the square.
   Full step-by-step solution

   Step 1: The larger square has side length 12, so its vertices are at (0,0), (12,0), (12,12), and (0,12). Step 2: The circle is inscribed in the larger square, meaning the circle touches all four sides of the square. The diameter of the circle equals the side length of the square, so diameter = 12. The radius is half of that: radius = 12 / 2 = 6. Step 3: The smaller square is inscribed in the circle, so its vertices lie on the circle. The diagonal of the smaller square is the diameter of the circle, which is 12. Step 4: For any square, the diagonal = side * sqrt(2). Let s be the side length of the smaller square. Then s * sqrt(2) = 12, so s = 12 / sqrt(2) = 6 * sqrt(2). Step 5: The problem states that the diagonal of the smaller square equals the side length of the larger square, which is 12. This matches our calculation: diagonal = 12 = side of larger square. Step 6: The radius of the circle is 6. The answer is 6.