How to Solve Quadratic Equations: Step-by-Step Guide

Solving quadratic equations can be a crucial skill in mathematics, especially in algebra. Let’s take a closer look at the equation you provided: $5x^2 + 75x = 0$. This equation is a classic example of a quadratic equation, which generally has the form $ax^2 + bx + c = 0$. In our case, we identify $a = 5$, $b = 75$, and $c = 0$.

### Step 1: Factor the Equation
The first step in solving this quadratic equation is to factor it. Notice that both terms on the left side of the equation share a common factor of $x$. We can factor out $x$:

$$5x^2 + 75x = x(5x + 75) = 0$$

### Step 2: Set Each Factor to Zero
Now that we have factored the equation, we can set each factor equal to zero. This is based on the property that if a product of factors equals zero, at least one of the factors must also equal zero. Therefore, we set:

1. $x = 0$
2. $5x + 75 = 0$

### Step 3: Solve Each Equation
The first equation is straightforward; if $x = 0$, that’s one solution. For the second equation, we need to solve for $x$:

$$5x + 75 = 0$$

Subtract 75 from both sides:
$$5x = -75$$

Now, divide by 5:
$$x = -15$$

### Step 4: Conclusion
In conclusion, the two solutions to the quadratic equation $5x^2 + 75x = 0$ are:
- $x = 0$
- $x = -15$

### Real-World Applications
Understanding how to solve quadratic equations is essential, as it applies to various real-world scenarios, such as calculating areas, optimizing profits in business, and analyzing projectile motion in physics. Quadratic equations can model numerous situations, making this skill not only academically important but also practically relevant.

### Additional Tips
When working with quadratic equations, always look for the possibility of factoring first. If factoring is difficult, you can also use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula provides a way to find the roots of any quadratic equation directly, regardless of whether it can be factored easily.

### Practice Problems
Try solving these quadratic equations:
1. $2x^2 + 8x = 0$
2. $x^2 - 4x - 12 = 0$
3. $3x^2 + 18x = 0$

By practicing, you'll become more confident in identifying and solving quadratic equations effectively.