What is the positive solution to the equation 4x^2 + 12x = 135
The Correct Answer and Explanation is :
To find the positive solution to the equation ( 4x^2 + 12x = 135 ), we can follow these steps:
- Rearrange the Equation: First, we want to set the equation to zero by moving 135 to the left side:
[
4x^2 + 12x – 135 = 0
] - Use the Quadratic Formula: The quadratic formula is given by:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In our equation, ( a = 4 ), ( b = 12 ), and ( c = -135 ). - Calculate the Discriminant: We start by calculating the discriminant (( b^2 – 4ac )):
[
b^2 = 12^2 = 144
]
[
4ac = 4 \cdot 4 \cdot (-135) = -2160
]
[
\text{Discriminant} = 144 – (-2160) = 144 + 2160 = 2304
] - Find the Roots: Now we can substitute into the quadratic formula:
[
x = \frac{-12 \pm \sqrt{2304}}{2 \cdot 4}
]
First, calculate ( \sqrt{2304} ):
[
\sqrt{2304} = 48
]
Substitute back into the formula:
[
x = \frac{-12 \pm 48}{8}
]
This gives us two possible solutions:
[
x = \frac{36}{8} = 4.5 \quad \text{and} \quad x = \frac{-60}{8} = -7.5
] - Select the Positive Solution: Since we are looking for the positive solution, we take:
[
x = 4.5
]
Conclusion
The positive solution to the equation ( 4x^2 + 12x = 135 ) is ( x = 4.5 ). This solution was derived using the quadratic formula, which is a reliable method for solving quadratic equations. The steps involved rearranging the equation, calculating the discriminant to check for real solutions, and then applying the quadratic formula to find the roots. By isolating the positive root, we determined the required solution effectively.