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To solve the quadratic equation [tex]\( x^2 - 12x + 45 = 0 \)[/tex], follow these steps:
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, we identify:
[tex]\[ a = 1, \quad b = -12, \quad c = 45 \][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 45 \)[/tex]:
[tex]\[ \Delta = (-12)^2 - 4 \times 1 \times 45 = 144 - 180 = -36 \][/tex]
Since the discriminant is negative, the quadratic equation has two complex roots.
3. Apply the quadratic formula:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since the discriminant [tex]\( \Delta = -36 \)[/tex], we have:
[tex]\[ \sqrt{\Delta} = \sqrt{-36} = 6i \][/tex]
(Here, [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex]).
Substitute the values of [tex]\( b = -12 \)[/tex], [tex]\( \sqrt{\Delta} = 6i \)[/tex], and [tex]\( a = 1 \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{12 \pm 6i}{2} \][/tex]
4. Simplify each root:
Simplify the expression for each root:
[tex]\[ x = \frac{12 + 6i}{2} \quad \text{and} \quad x = \frac{12 - 6i}{2} \][/tex]
This gives:
[tex]\[ x = 6 + 3i \quad \text{and} \quad x = 6 - 3i \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 12x + 45 = 0 \)[/tex] are [tex]\( x = 6 + 3i \)[/tex] and [tex]\( x = 6 - 3i \)[/tex].
The correct answer from the given options is:
[tex]\[ \boxed{B. \, x = 6 \pm 3i} \][/tex]
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, we identify:
[tex]\[ a = 1, \quad b = -12, \quad c = 45 \][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 45 \)[/tex]:
[tex]\[ \Delta = (-12)^2 - 4 \times 1 \times 45 = 144 - 180 = -36 \][/tex]
Since the discriminant is negative, the quadratic equation has two complex roots.
3. Apply the quadratic formula:
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since the discriminant [tex]\( \Delta = -36 \)[/tex], we have:
[tex]\[ \sqrt{\Delta} = \sqrt{-36} = 6i \][/tex]
(Here, [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex]).
Substitute the values of [tex]\( b = -12 \)[/tex], [tex]\( \sqrt{\Delta} = 6i \)[/tex], and [tex]\( a = 1 \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{12 \pm 6i}{2} \][/tex]
4. Simplify each root:
Simplify the expression for each root:
[tex]\[ x = \frac{12 + 6i}{2} \quad \text{and} \quad x = \frac{12 - 6i}{2} \][/tex]
This gives:
[tex]\[ x = 6 + 3i \quad \text{and} \quad x = 6 - 3i \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 12x + 45 = 0 \)[/tex] are [tex]\( x = 6 + 3i \)[/tex] and [tex]\( x = 6 - 3i \)[/tex].
The correct answer from the given options is:
[tex]\[ \boxed{B. \, x = 6 \pm 3i} \][/tex]
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