The solution to x2 – 10x = 24 is
The Correct Answer and Explanation is :
The correct answer is: ( x = 12 ) and ( x = -2 ).
To solve the quadratic equation ( x^2 – 10x = 24 ), we first rearrange it into standard form by moving all terms to one side of the equation:
[
x^2 – 10x – 24 = 0
]
Next, we can use the quadratic formula to find the solutions. The quadratic formula states that for any equation of the form ( ax^2 + bx + c = 0 ), the solutions for ( x ) can be found using:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In our case, ( a = 1 ), ( b = -10 ), and ( c = -24 ). We will first calculate the discriminant (( b^2 – 4ac )):
[
b^2 – 4ac = (-10)^2 – 4 \cdot 1 \cdot (-24)
]
[
= 100 + 96
]
[
= 196
]
Since the discriminant is positive, we will have two distinct real solutions. Now, we can substitute our values into the quadratic formula:
[
x = \frac{-(-10) \pm \sqrt{196}}{2 \cdot 1}
]
[
= \frac{10 \pm 14}{2}
]
This gives us two possible solutions:
- ( x = \frac{10 + 14}{2} = \frac{24}{2} = 12 )
- ( x = \frac{10 – 14}{2} = \frac{-4}{2} = -2 )
Thus, the solutions to the equation ( x^2 – 10x = 24 ) are ( x = 12 ) and ( x = -2 ).
In conclusion, we have determined that the quadratic equation ( x^2 – 10x – 24 = 0 ) has two solutions: ( x = 12 ) and ( x = -2 ). This process illustrates how to rearrange a quadratic equation into standard form, apply the quadratic formula, and interpret the discriminant to determine the nature of the solutions. Understanding these concepts is crucial for solving quadratic equations effectively, as they frequently appear in various fields of study, including mathematics, physics, and engineering.