Solving Quadratic Equations: Find The Other Solution

Let's dive into the world of quadratic equations! In this article, we'll explore how to solve a quadratic equation when one of the solutions is already known. We'll use the given equation $3x^2 + 6x - 24 = 0$ as our example, where one solution is $x = -4$, and we need to find the other solution. So, buckle up and let's get started!

Understanding Quadratic Equations

First, let's briefly discuss what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The solutions to a quadratic equation are also known as its roots or zeros.

In our case, the given quadratic equation is $3x^2 + 6x - 24 = 0$. Notice that $a = 3$, $b = 6$, and $c = -24$. We are already given that one of the solutions to this equation is $x = -4$. Our mission is to find the other solution. There are a couple of ways we can approach this problem, and we'll explore both.

Method 1: Factoring and Using the Given Root

One way to solve this is by factoring the quadratic equation. Factoring involves expressing the quadratic expression as a product of two linear expressions. Since we know that $x = -4$ is a solution, we know that $(x + 4)$ must be one of the factors. Here's how we can proceed:

Step 1: Factor out the common factor, if any.

Looking at the equation $3x^2 + 6x - 24 = 0$, we can see that each term is divisible by 3. Factoring out 3 simplifies the equation to:

$3(x^2 + 2x - 8) = 0$

Step 2: Factor the quadratic expression.

Now, we need to factor the quadratic expression $x^2 + 2x - 8$. We are looking for two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. So, we can write the quadratic expression as:

$(x + 4)(x - 2) = 0$

Step 3: Write the factored form of the equation.

Putting it all together, the factored form of the original equation is:

$3(x + 4)(x - 2) = 0$

Step 4: Find the solutions.

To find the solutions, we set each factor equal to zero:

$x + 4 = 0$ or $x - 2 = 0$

Solving for $x$ in each case gives us:

$x = -4$ or $x = 2$

So, the two solutions are $x = -4$ and $x = 2$. Since we were given that one solution is $x = -4$, the other solution must be $x = 2$.

Method 2: Using Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. For a quadratic equation $ax^2 + bx + c = 0$ with roots $x_1$ and $x_2$, Vieta's formulas state:

$x_1 + x_2 = -\frac{b}{a}$ $x_1 \, x_2 = \frac{c}{a}$

In our case, the equation is $3x^2 + 6x - 24 = 0$, so $a = 3$, $b = 6$, and $c = -24$. We know one solution is $x_1 = -4$. Let's call the other solution $x_2$. We can use either of Vieta's formulas to find $x_2$. Let's use the sum of the roots formula:

Step 1: Apply Vieta's formula for the sum of roots.

$x_1 + x_2 = -\frac{b}{a}$

Substitute the known values:

$-4 + x_2 = -\frac{6}{3}$

Step 2: Simplify and solve for the unknown root.

$-4 + x_2 = -2$

Add 4 to both sides:

$x_2 = -2 + 4$

$x_2 = 2$

Thus, the other solution is $x = 2$.

Method 3: Using the Quadratic Formula

For any quadratic equation $ax^2 + bx + c = 0$, the solutions can be found using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our case, the equation is $3x^2 + 6x - 24 = 0$, so $a = 3$, $b = 6$, and $c = -24$. Let's plug these values into the quadratic formula:

Step 1: Substitute the values into the quadratic formula.

$x = \frac{-6 \pm \sqrt{6^2 - 4(3)(-24)}}{2(3)}$

Step 2: Simplify the expression.

$x = \frac{-6 \pm \sqrt{36 + 288}}{6}$

$x = \frac{-6 \pm \sqrt{324}}{6}$

$x = \frac{-6 \pm 18}{6}$

Step 3: Find the two solutions.

Now we have two possible solutions:

$x_1 = \frac{-6 + 18}{6} = \frac{12}{6} = 2$

$x_2 = \frac{-6 - 18}{6} = \frac{-24}{6} = -4$

So, the two solutions are $x = 2$ and $x = -4$. Since we were given that one solution is $x = -4$, the other solution must be $x = 2$.

Conclusion

We have successfully found the other solution to the quadratic equation $3x^2 + 6x - 24 = 0$, given that one solution is $x = -4$. We explored three different methods: factoring, using Vieta's formulas, and applying the quadratic formula. All three methods led us to the same answer: the other solution is $x = 2$. Understanding these methods will give you a solid foundation for solving various quadratic equations and related problems. Keep practicing, and you'll become a quadratic equation solving pro in no time! Don't forget to practice and explore different types of quadratic equations to master this skill.

For more information on quadratic equations, you can visit Khan Academy's Quadratic Equations Section.