The solution to x2 – 10x = 24 is
✔ 12 or –2
The solution to 2×2 – 11 = 87 is
✔ 7 or –7
The solution to 3×2 – 12x + 24 = 0 is
✔ 2 + 2i or 2 – 2i
The Correct Answer and Explanation is:
Let’s solve the equations one by one and provide the detailed explanation.
1. Solution to (x^2 – 10x = 24)
Rearrange the equation to standard quadratic form:
[
x^2 – 10x – 24 = 0
]
Now, we can solve this quadratic equation using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
For the equation (x^2 – 10x – 24 = 0), the coefficients are (a = 1), (b = -10), and (c = -24).
Substitute these values into the quadratic formula:
[
x = \frac{-(-10) \pm \sqrt{(-10)^2 – 4(1)(-24)}}{2(1)}
]
[
x = \frac{10 \pm \sqrt{100 + 96}}{2}
]
[
x = \frac{10 \pm \sqrt{196}}{2}
]
[
x = \frac{10 \pm 14}{2}
]
Thus, the two possible solutions are:
[
x = \frac{10 + 14}{2} = \frac{24}{2} = 12
]
and
[
x = \frac{10 – 14}{2} = \frac{-4}{2} = -2
]
So, the solutions are 12 and -2.
2. Solution to (2x^2 – 11 = 87)
Rearrange the equation:
[
2x^2 – 11 = 87 \quad \Rightarrow \quad 2x^2 = 87 + 11 \quad \Rightarrow \quad 2x^2 = 98
]
Now, divide both sides by 2:
[
x^2 = \frac{98}{2} = 49
]
Taking the square root of both sides:
[
x = \pm \sqrt{49} = \pm 7
]
So, the solutions are 7 and -7.
3. Solution to (3x^2 – 12x + 24 = 0)
We can solve this quadratic equation using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
For the equation (3x^2 – 12x + 24 = 0), the coefficients are (a = 3), (b = -12), and (c = 24).
Substitute these values into the quadratic formula:
[
x = \frac{-(-12) \pm \sqrt{(-12)^2 – 4(3)(24)}}{2(3)}
]
[
x = \frac{12 \pm \sqrt{144 – 288}}{6}
]
[
x = \frac{12 \pm \sqrt{-144}}{6}
]
Since the discriminant ((\sqrt{-144})) is negative, the solutions will be complex numbers. We can express the square root of (-144) as:
[
\sqrt{-144} = 12i
]
Thus, the solutions are:
[
x = \frac{12 \pm 12i}{6}
]
Simplifying:
[
x = \frac{12}{6} \pm \frac{12i}{6}
]
[
x = 2 \pm 2i
]
So, the solutions are (2 + 2i) and (2 – 2i).
Summary of Solutions:
- (x^2 – 10x = 24) → 12 or -2
- (2x^2 – 11 = 87) → 7 or -7
- (3x^2 – 12x + 24 = 0) → 2 + 2i or 2 – 2i
These solutions are derived through solving each quadratic equation either by factoring, completing the square, or using the quadratic formula.