Radical Equations

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Answer Key & Explanations

Radical Equations · Grade 9 · Worksheet 1

1. A right triangle is positioned on a coordinate plane with vertices at (0,0), (12,0), and (0,5). A circle is drawn such that it is tangent to the hypotenuse and to both legs of the triangle. What is the radius of this inscribed circle? Answer: 2 Solution: Identify the side lengths of the triangle. The legs are along the axes, so their lengths are 12 and 5. The hypotenuse length is found using the Pythagorean theorem: sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13.
   Full step-by-step solution

   Step 1: Identify the side lengths of the triangle. The legs are along the axes, so their lengths are 12 and 5. The hypotenuse length is found using the Pythagorean theorem: sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13. Step 2: The formula for the radius (r) of a circle inscribed in a triangle is r = A / s, where A is the area and s is the semiperimeter. Step 3: Calculate the area (A) of the triangle. A = (1/2) * base * height = (1/2) * 12 * 5 = 30. Step 4: Calculate the semiperimeter (s). s = (a + b + c)/2 = (12 + 5 + 13)/2 = 30/2 = 15. Step 5: Calculate the radius. r = A / s = 30 / 15 = 2. The radius of the inscribed circle is 2.

2. √(2x + 5) - 3 = 0 Answer: 2 Solution: √(2x + 5) - 3 = 0 Isolate the square root term. Add 3 to both sides of the equation: √(2x + 5) = 3 Square both sides to remove the square root. (√(2x + 5))^2 = (3)^2 2x + 5 = 9 Solve for x.
   Full step-by-step solution

   We start with the equation: √(2x + 5) - 3 = 0 Step 1: Isolate the square root term. Add 3 to both sides of the equation: √(2x + 5) = 3 Step 2: Square both sides to remove the square root. (√(2x + 5))^2 = (3)^2 This simplifies to: 2x + 5 = 9 Step 3: Solve for x. Subtract 5 from both sides: 2x = 4 Divide both sides by 2: x = 2 Step 4: Check for extraneous solutions by substituting back into the original equation. Original equation: √(2x + 5) - 3 = 0 Substitute x = 2: √(2*2 + 5) - 3 = √(4 + 5) - 3 = √9 - 3 = 3 - 3 = 0 This is correct, so x = 2 is valid. Final answer: 2

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (15,0), and (0,8). A circle is inscribed inside this triangle such that it touches all three sides. What is the radius of this inscribed circle? Answer: 3 Solution: Identify the triangle sides. The legs are 15 units (horizontal) and 8 units (vertical). The hypotenuse can be found using Pythagorean theorem: sqrt(15^2 + 8^2) = sqrt(225 + 64) = sqrt(289) = 17 units.
   Full step-by-step solution

   Step 1: Identify the triangle sides. The legs are 15 units (horizontal) and 8 units (vertical). The hypotenuse can be found using Pythagorean theorem: sqrt(15^2 + 8^2) = sqrt(225 + 64) = sqrt(289) = 17 units. Step 2: For a right triangle with legs a and b, and hypotenuse c, the radius r of the inscribed circle is given by the formula: r = (a + b - c)/2. Step 3: Substitute the values: a = 15, b = 8, c = 17 r = (15 + 8 - 17)/2 = (23 - 17)/2 = 6/2 = 3 Step 4: The radius of the inscribed circle is 3 units. The answer is 3.

4. Kaia is a marine engineer testing a new underwater drone. The drone's depth d (in meters) below the surface is modeled by the function d(t) = sqrt(7t + 51), where t is the time in seconds since the drone started descending. If the drone reaches a depth of exactly 11 meters, how many seconds has it been descending? Answer: 10 Solution: Write the equation with the given depth: sqrt(7t + 51) = 11 Square both sides to eliminate the square root: (sqrt(7t + 51))^2 = 11^2 Simplify: 7t + 51 = 121 Subtract 51 from both sides: 7t = 70 Divide both sides by 7: t = 10 Check by substituting t = 10 back into the original equation:…
   Full step-by-step solution

   Step 1: Write the equation with the given depth: sqrt(7t + 51) = 11 Step 2: Square both sides to eliminate the square root: (sqrt(7t + 51))^2 = 11^2 Step 3: Simplify: 7t + 51 = 121 Step 4: Subtract 51 from both sides: 7t = 70 Step 5: Divide both sides by 7: t = 10 Step 6: Check by substituting t = 10 back into the original equation: sqrt(7(10) + 51) = sqrt(70 + 51) = sqrt(121) = 11. The solution is valid. The drone has been descending for 10 seconds.

5. √(5x - 19) - 3 = 0 Answer: x = 28/5 Solution: Add 3 to both sides to isolate the radical: √(5x - 19) = 3 Square both sides: (√(5x - 19))² = 3² Simplify: 5x - 19 = 9 Add 19 to both sides: 5x = 28 Divide both sides by 5: x = 28/5 Check the solution in the original equation: √(5(28/5) - 19) - 3 = √(28 - 19) - 3 = √9 - 3 = 3 - 3 = 0 The…
   Full step-by-step solution

   Step 1: Add 3 to both sides to isolate the radical: √(5x - 19) = 3 Step 2: Square both sides: (√(5x - 19))² = 3² Step 3: Simplify: 5x - 19 = 9 Step 4: Add 19 to both sides: 5x = 28 Step 5: Divide both sides by 5: x = 28/5 Step 6: Check the solution in the original equation: √(5(28/5) - 19) - 3 = √(28 - 19) - 3 = √9 - 3 = 3 - 3 = 0 The solution checks out. The answer is x = 28/5.

6. √(4x - 7) + 5 = 12 Answer: 14 Solution: Isolate the radical expression by subtracting 5 from both sides √(4x - 7) + 5 - 5 = 12 - 5 √(4x - 7) = 7 (√(4x - 7))² = 7² 4x - 7 = 49 4x - 7 = 49 4x = 49 + 7 4x = 56 x = 56 ÷ 4 x = 14 √(4(14) - 7) + 5 = √(56 - 7) + 5 = √49 + 5 = 7 + 5 = 12 ✓ The answer is 14.
   Full step-by-step solution

   Step 1: Isolate the radical expression by subtracting 5 from both sides √(4x - 7) + 5 - 5 = 12 - 5 √(4x - 7) = 7 Step 2: Square both sides to eliminate the square root (√(4x - 7))² = 7² 4x - 7 = 49 Step 3: Solve the resulting linear equation 4x - 7 = 49 4x = 49 + 7 4x = 56 x = 56 ÷ 4 x = 14 Step 4: Check the solution in the original equation √(4(14) - 7) + 5 = √(56 - 7) + 5 = √49 + 5 = 7 + 5 = 12 ✓ The answer is 14.

7. A square is drawn on a coordinate plane with vertices at (0,0), (7,0), (7,7), and (0,7). A circle is inscribed inside the square such that it touches all four sides. A right triangle is drawn inside the circle with its hypotenuse as the diameter of the circle and one leg along the horizontal diameter from the center to the right edge. If the length of the horizontal leg is 2 units, what is the length of the other leg? Write your answer in the form a√b, where a and b are integers. Answer: √45 or 3√5 Solution: The square has side length 7. The inscribed circle touches all four sides, so its diameter equals the side length of the square. The triangle has its hypotenuse as the diameter of the circle.
   Full step-by-step solution

   Step 1: The square has side length 7. The inscribed circle touches all four sides, so its diameter equals the side length of the square. Therefore, the diameter of the circle is 7 units, and the radius is 7/2 = 3.5 units. Step 2: The triangle has its hypotenuse as the diameter of the circle. So the hypotenuse length is 7 units. Step 3: One leg of the right triangle is given as 2 units. Let the other leg be x. Step 4: Apply the Pythagorean theorem: (leg 1)^2 + (leg 2)^2 = (hypotenuse)^2 2^2 + x^2 = 7^2 4 + x^2 = 49 x^2 = 49 - 4 x^2 = 45 x = √45 Step 5: Simplify √45 = √(9 × 5) = 3√5 The length of the other leg is √45 or 3√5 units.

8. √(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