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Answer :
To find the width of the pool, we need to solve the quadratic equation [tex]\(x^2 + 2x - 120 = 0\)[/tex].
Step 1: Identify the coefficients
The quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex]. From the equation:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -120\)[/tex]
Step 2: Calculate the discriminant
The discriminant ([tex]\(D\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[D = b^2 - 4ac\][/tex]
Plugging in the values:
[tex]\[D = 2^2 - 4 \times 1 \times (-120)\][/tex]
[tex]\[D = 4 + 480\][/tex]
[tex]\[D = 484\][/tex]
Step 3: Solve for the roots
The roots of the quadratic equation can be calculated using the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{D}}{2a}\][/tex]
With our values:
[tex]\[x = \frac{-2 \pm \sqrt{484}}{2 \times 1}\][/tex]
Step 4: Calculate the two possible solutions
Since [tex]\(\sqrt{484} = 22\)[/tex], we find the roots:
- [tex]\(x_1 = \frac{-2 + 22}{2} = \frac{20}{2} = 10\)[/tex]
- [tex]\(x_2 = \frac{-2 - 22}{2} = \frac{-24}{2} = -12\)[/tex]
Step 5: Choose the positive root
Since the width of the pool cannot be negative, the valid solution is the positive root:
- The width of the pool, [tex]\(x\)[/tex], is 10 feet.
Therefore, the value of [tex]\(x\)[/tex], the width of the pool, is [tex]\(10\)[/tex] feet.
Step 1: Identify the coefficients
The quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex]. From the equation:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -120\)[/tex]
Step 2: Calculate the discriminant
The discriminant ([tex]\(D\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[D = b^2 - 4ac\][/tex]
Plugging in the values:
[tex]\[D = 2^2 - 4 \times 1 \times (-120)\][/tex]
[tex]\[D = 4 + 480\][/tex]
[tex]\[D = 484\][/tex]
Step 3: Solve for the roots
The roots of the quadratic equation can be calculated using the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{D}}{2a}\][/tex]
With our values:
[tex]\[x = \frac{-2 \pm \sqrt{484}}{2 \times 1}\][/tex]
Step 4: Calculate the two possible solutions
Since [tex]\(\sqrt{484} = 22\)[/tex], we find the roots:
- [tex]\(x_1 = \frac{-2 + 22}{2} = \frac{20}{2} = 10\)[/tex]
- [tex]\(x_2 = \frac{-2 - 22}{2} = \frac{-24}{2} = -12\)[/tex]
Step 5: Choose the positive root
Since the width of the pool cannot be negative, the valid solution is the positive root:
- The width of the pool, [tex]\(x\)[/tex], is 10 feet.
Therefore, the value of [tex]\(x\)[/tex], the width of the pool, is [tex]\(10\)[/tex] feet.
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