Answer :
- Rewrite the given quadratic equation in standard form: $2x^2 + 15x - 108 = 0$.
- Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- Calculate the discriminant: $D = b^2 - 4ac = 1089$.
- Find the two solutions: $x_1 = 4.5$ and $x_2 = -12$. The solutions are $\boxed{x=-12 \text{ and } x=4.5}$.
### Explanation
1. Problem Analysis
We are given the quadratic equation $2x^2 + 15x = 108$. Our goal is to find the solutions for $x$.
2. Rewriting the Equation
First, we need to rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$. Subtracting 108 from both sides, we get $2x^2 + 15x - 108 = 0$.
3. Applying the Quadratic Formula
Now, we can use the quadratic formula to solve for $x$. The quadratic formula is given by $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$, where $a=2$, $b=15$, and $c=-108$.
4. Calculating the Discriminant
Let's calculate the discriminant, which is the part under the square root: $D = b^2 - 4ac = 15^2 - 4(2)(-108) = 225 + 864 = 1089$.
5. Substituting into the Formula
Now, we can plug the values of $a$, $b$, $c$, and $D$ into the quadratic formula:
$x = \\frac{-15 \\pm \\sqrt{1089}}{2(2)} = \\frac{-15 \\pm 33}{4}$.
6. Finding the Solutions
We have two possible solutions for $x$:
$x_1 = \\frac{-15 + 33}{4} = \\frac{18}{4} = 4.5$
$x_2 = \\frac{-15 - 33}{4} = \\frac{-48}{4} = -12$
7. Final Answer
Therefore, the solutions are $x = 4.5$ and $x = -12$.
### Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For instance, if you're designing a bridge, you might use a quadratic equation to model the arch's shape and ensure it can withstand specific loads. Understanding how to solve these equations is crucial for making accurate predictions and informed decisions in these scenarios.
- Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- Calculate the discriminant: $D = b^2 - 4ac = 1089$.
- Find the two solutions: $x_1 = 4.5$ and $x_2 = -12$. The solutions are $\boxed{x=-12 \text{ and } x=4.5}$.
### Explanation
1. Problem Analysis
We are given the quadratic equation $2x^2 + 15x = 108$. Our goal is to find the solutions for $x$.
2. Rewriting the Equation
First, we need to rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$. Subtracting 108 from both sides, we get $2x^2 + 15x - 108 = 0$.
3. Applying the Quadratic Formula
Now, we can use the quadratic formula to solve for $x$. The quadratic formula is given by $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$, where $a=2$, $b=15$, and $c=-108$.
4. Calculating the Discriminant
Let's calculate the discriminant, which is the part under the square root: $D = b^2 - 4ac = 15^2 - 4(2)(-108) = 225 + 864 = 1089$.
5. Substituting into the Formula
Now, we can plug the values of $a$, $b$, $c$, and $D$ into the quadratic formula:
$x = \\frac{-15 \\pm \\sqrt{1089}}{2(2)} = \\frac{-15 \\pm 33}{4}$.
6. Finding the Solutions
We have two possible solutions for $x$:
$x_1 = \\frac{-15 + 33}{4} = \\frac{18}{4} = 4.5$
$x_2 = \\frac{-15 - 33}{4} = \\frac{-48}{4} = -12$
7. Final Answer
Therefore, the solutions are $x = 4.5$ and $x = -12$.
### Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For instance, if you're designing a bridge, you might use a quadratic equation to model the arch's shape and ensure it can withstand specific loads. Understanding how to solve these equations is crucial for making accurate predictions and informed decisions in these scenarios.