Answer :
Sure! Let's solve each of the given quadratic equations step-by-step:
1. Equation: [tex]\(2x^2 - 32 = 0\)[/tex]
- Step 1: Add 32 to both sides to isolate the quadratic term.
[tex]\[2x^2 = 32\][/tex]
- Step 2: Divide both sides by 2.
[tex]\[x^2 = 16\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm4\][/tex]
- Solution set: [tex]\([-4, 4]\)[/tex]
2. Equation: [tex]\(4x^2 - 100 = 0\)[/tex]
- Step 1: Add 100 to both sides to isolate the quadratic term.
[tex]\[4x^2 = 100\][/tex]
- Step 2: Divide both sides by 4.
[tex]\[x^2 = 25\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm5\][/tex]
- Solution set: [tex]\([-5, 5]\)[/tex]
3. Equation: [tex]\(x^2 - 55 = 9\)[/tex]
- Step 1: Add 55 to both sides to isolate the quadratic term.
[tex]\[x^2 = 64\][/tex]
- Step 2: Take the square root of both sides.
[tex]\[x = \pm8\][/tex]
- Solution set: [tex]\([-8, 8]\)[/tex]
4. Equation: [tex]\(x^2 - 140 = -19\)[/tex]
- Step 1: Add 140 to both sides to isolate the quadratic term.
[tex]\[x^2 = 121\][/tex]
- Step 2: Take the square root of both sides.
[tex]\[x = \pm11\][/tex]
- Solution set: [tex]\([-11, 11]\)[/tex]
5. Equation: [tex]\(2x^2 - 18 = 0\)[/tex]
- Step 1: Add 18 to both sides to isolate the quadratic term.
[tex]\[2x^2 = 18\][/tex]
- Step 2: Divide both sides by 2.
[tex]\[x^2 = 9\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm3\][/tex]
- Solution set: [tex]\([-3, 3]\)[/tex]
To summarize, the solutions are:
1. [tex]\(2x^2 - 32 = 0\)[/tex] has solutions [tex]\([-4, 4]\)[/tex]
2. [tex]\(4x^2 - 100 = 0\)[/tex] has solutions [tex]\([-5, 5]\)[/tex]
3. [tex]\(x^2 - 55 = 9\)[/tex] has solutions [tex]\([-8, 8]\)[/tex]
4. [tex]\(x^2 - 140 = -19\)[/tex] has solutions [tex]\([-11, 11]\)[/tex]
5. [tex]\(2x^2 - 18 = 0\)[/tex] has solutions [tex]\([-3, 3]\)[/tex]
These are the detailed, step-by-step solutions for each equation.
1. Equation: [tex]\(2x^2 - 32 = 0\)[/tex]
- Step 1: Add 32 to both sides to isolate the quadratic term.
[tex]\[2x^2 = 32\][/tex]
- Step 2: Divide both sides by 2.
[tex]\[x^2 = 16\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm4\][/tex]
- Solution set: [tex]\([-4, 4]\)[/tex]
2. Equation: [tex]\(4x^2 - 100 = 0\)[/tex]
- Step 1: Add 100 to both sides to isolate the quadratic term.
[tex]\[4x^2 = 100\][/tex]
- Step 2: Divide both sides by 4.
[tex]\[x^2 = 25\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm5\][/tex]
- Solution set: [tex]\([-5, 5]\)[/tex]
3. Equation: [tex]\(x^2 - 55 = 9\)[/tex]
- Step 1: Add 55 to both sides to isolate the quadratic term.
[tex]\[x^2 = 64\][/tex]
- Step 2: Take the square root of both sides.
[tex]\[x = \pm8\][/tex]
- Solution set: [tex]\([-8, 8]\)[/tex]
4. Equation: [tex]\(x^2 - 140 = -19\)[/tex]
- Step 1: Add 140 to both sides to isolate the quadratic term.
[tex]\[x^2 = 121\][/tex]
- Step 2: Take the square root of both sides.
[tex]\[x = \pm11\][/tex]
- Solution set: [tex]\([-11, 11]\)[/tex]
5. Equation: [tex]\(2x^2 - 18 = 0\)[/tex]
- Step 1: Add 18 to both sides to isolate the quadratic term.
[tex]\[2x^2 = 18\][/tex]
- Step 2: Divide both sides by 2.
[tex]\[x^2 = 9\][/tex]
- Step 3: Take the square root of both sides.
[tex]\[x = \pm3\][/tex]
- Solution set: [tex]\([-3, 3]\)[/tex]
To summarize, the solutions are:
1. [tex]\(2x^2 - 32 = 0\)[/tex] has solutions [tex]\([-4, 4]\)[/tex]
2. [tex]\(4x^2 - 100 = 0\)[/tex] has solutions [tex]\([-5, 5]\)[/tex]
3. [tex]\(x^2 - 55 = 9\)[/tex] has solutions [tex]\([-8, 8]\)[/tex]
4. [tex]\(x^2 - 140 = -19\)[/tex] has solutions [tex]\([-11, 11]\)[/tex]
5. [tex]\(2x^2 - 18 = 0\)[/tex] has solutions [tex]\([-3, 3]\)[/tex]
These are the detailed, step-by-step solutions for each equation.
