Radical Equations

lessonbunny.com

Answer Key & Explanations

Radical Equations · Grade 9 · Worksheet 3

1. The sum of a number and its square root is 12. Find the number. Answer: 9 Solution: Let x be the number.
   Full step-by-step solution

   Step 1: Let x be the number. The problem states: x + sqrt(x) = 12 Step 2: Let u = sqrt(x), so u^2 = x Step 3: Substitute into the equation: u^2 + u = 12 Step 4: Rearrange to standard quadratic form: u^2 + u - 12 = 0 Step 5: Factor the quadratic: (u + 4)(u - 3) = 0 Step 6: Solve for u: u = -4 or u = 3 Step 7: Since u = sqrt(x) must be non-negative, discard u = -4 Step 8: Use u = 3 to find x: x = u^2 = 3^2 = 9 Step 9: Verify: 9 + sqrt(9) = 9 + 3 = 12 The answer is 9.

2. Charlotte is designing a suspension bridge for a science fair project. The parabolic main cable of her bridge is modeled by the equation y = sqrt(2x + 27), where y is the height of the cable in centimeters above the bridge deck at a horizontal distance x centimeters from the left tower. Charlotte wants to place a vertical support beam at the point where the cable is exactly 7 centimeters above the deck. At what horizontal distance x from the left tower should she place the support beam? Answer: 11 Solution: Write the given equation with the known height: sqrt(2x + 27) = 7 Square both sides to eliminate the square root: (sqrt(2x + 27))^2 = 7^2 Simplify: 2x + 27 = 49 Subtract 27 from both sides: 2x = 49 - 27, so 2x = 22 Divide both sides by 2: x = 11 Check the solution by substituting x = 11 into the…
   Full step-by-step solution

   Step 1: Write the given equation with the known height: sqrt(2x + 27) = 7 Step 2: Square both sides to eliminate the square root: (sqrt(2x + 27))^2 = 7^2 Step 3: Simplify: 2x + 27 = 49 Step 4: Subtract 27 from both sides: 2x = 49 - 27, so 2x = 22 Step 5: Divide both sides by 2: x = 11 Step 6: Check the solution by substituting x = 11 into the original equation: sqrt(2(11) + 27) = sqrt(22 + 27) = sqrt(49) = 7. This matches the given height, so the solution is valid. The horizontal distance from the left tower is 11 centimeters.

3. √(3x + 7) = 4 Answer: 3 Solution: √(3x + 7) = 4 Since the square root is already isolated, square both sides to remove the square root. (√(3x + 7))^2 = 4^2 Simplify both sides.
   Full step-by-step solution

   We are solving the equation: √(3x + 7) = 4 Step 1: Since the square root is already isolated, square both sides to remove the square root. (√(3x + 7))^2 = 4^2 Step 2: Simplify both sides. Left side: (√(3x + 7))^2 = 3x + 7 Right side: 4^2 = 16 So we have: 3x + 7 = 16 Step 3: Subtract 7 from both sides to isolate the term with x. 3x + 7 - 7 = 16 - 7 3x = 9 Step 4: Divide both sides by 3 to solve for x. 3x / 3 = 9 / 3 x = 3 Step 5: Check the solution in the original equation. Original: √(3x + 7) = 4 Substitute x = 3: √(3*3 + 7) = √(9 + 7) = √16 = 4 This matches the right-hand side, so the solution is correct. Final answer: x = 3

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (21,0), and (0,28). A square is inscribed in this triangle such that one side of the square lies along the base of the triangle (the side from (0,0) to (21,0)), and the opposite side of the square touches the hypotenuse. What is the side length of this inscribed square? Answer: 12 Solution: The large triangle has base 21 and height 28. The hypotenuse runs from (0,28) to (21,0). Let the side length of the square be s.
   Full step-by-step solution

   Step 1: The large triangle has base 21 and height 28. The hypotenuse runs from (0,28) to (21,0). Step 2: Let the side length of the square be s. The square sits on the base, so its top side is at height s (since the square's side length is vertical distance from the base). Step 3: The small triangle above the square is similar to the large triangle. Its height is 28 - s, and its base is the same as the top side of the square, which is s. Step 4: By similarity of triangles, the ratio of base to height is the same: base/height = 21/28 = 3/4. Step 5: For the small triangle: base/height = s/(28 - s) = 3/4. Step 6: Solve the proportion: 4s = 3(28 - s) -> 4s = 84 - 3s -> 7s = 84 -> s = 12. Step 7: The side length of the inscribed square is 12 units. The answer is 12.

5. 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 find the dimensions of Liam's garden. Answer: width = 4.5 meters, length = 12 meters 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 Given area = 54 square meters: (2w + 3) \times w = 54 2w^2 + 3w = 54 Subtract 54 from both sides: 2w^2 + 3w - 54 = 0 For \( a w^2 + b w…
   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: For \( a w^2 + b w + c = 0 \), \( a = 2 \), \( b = 3 \), \( c = -54 \) \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ w = \frac{-3 \pm \sqrt{3^2 - 4(2)(-54)}}{2 \times 2} \] \[ w = \frac{-3 \pm \sqrt{9 + 432}}{4} \] \[ w = \frac{-3 \pm \sqrt{441}}{4} \] \[ w = \frac{-3 \pm 21}{4} \] --- **Step 5: Two possible solutions** First: \( w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \) Second: \( w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \) Width cannot be negative, so \( w = 4.5 \) meters. --- **Step 6: Find length** \[ \text{length} = 2w + 3 = 2(4.5) + 3 = 9 + 3 = 12 \ \text{meters} \] --- **Step 7: Check** Area = \( 12 \times 4.5 = 54 \) square meters. Length (12) is indeed 3 more than twice the width (twice 4.5 = 9, plus 3 = 12). --- **Final answer:** Width = 4.5 meters, Length = 12 meters

6. Matiu is an engineer testing the stability of a new bridge design. The vertical displacement y (in millimeters) of the bridge at a horizontal distance x (in meters) from the left support is given by the equation y = √(4x + 12). During a stress test, Matiu measures the displacement at a certain point to be exactly 8 millimeters. What is the horizontal distance x (in meters) from the left support to that point? Answer: 13 Solution: Set up the equation using the given displacement: √(4x + 12) = 8 Square both sides to eliminate the square root: (√(4x + 12))² = 8² Simplify: 4x + 12 = 64 Subtract 12 from both sides: 4x = 52 Divide both sides by 4: x = 13 Check the solution by plugging x = 13 back into the original equation:…
   Full step-by-step solution

   Step 1: Set up the equation using the given displacement: √(4x + 12) = 8 Step 2: Square both sides to eliminate the square root: (√(4x + 12))² = 8² Step 3: Simplify: 4x + 12 = 64 Step 4: Subtract 12 from both sides: 4x = 52 Step 5: Divide both sides by 4: x = 13 Step 6: Check the solution by plugging x = 13 back into the original equation: √(4(13) + 12) = √(52 + 12) = √64 = 8. The solution is valid. The horizontal distance from the left support is 13 meters.

7. Hana is a materials scientist testing a new alloy. The strength S (in megapascals) of the alloy at temperature T (in degrees Celsius) is modeled by the equation S = √(4T + 68). If the alloy's strength is measured to be exactly 12 megapascals during an experiment, what is the temperature T in degrees Celsius? Answer: 19 Solution: Write the equation with the given strength: √(4T + 68) = 12 Square both sides to eliminate the square root: (√(4T + 68))² = 12² Simplify: 4T + 68 = 144 Subtract 68 from both sides: 4T = 76 Divide both sides by 4: T = 19 Check by substituting T = 19 back into the original equation: √(4(19) + 68)…
   Full step-by-step solution

   Step 1: Write the equation with the given strength: √(4T + 68) = 12 Step 2: Square both sides to eliminate the square root: (√(4T + 68))² = 12² Step 3: Simplify: 4T + 68 = 144 Step 4: Subtract 68 from both sides: 4T = 76 Step 5: Divide both sides by 4: T = 19 Step 6: Check by substituting T = 19 back into the original equation: √(4(19) + 68) = √(76 + 68) = √144 = 12. The solution is valid. The temperature is 19 degrees Celsius.