Solving Quadratic Equations With The Quadratic Formula Find The Values Of X

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. In this article, we will explore how to use the quadratic formula to find the solutions (also called roots) of a given quadratic equation. Specifically, we will tackle the equation $x^2 + 20 = 2x$ and determine the values of $x$ that satisfy it. Understanding the quadratic formula is crucial for anyone studying algebra and beyond, as it provides a reliable method for solving equations that cannot be easily factored.

Understanding the Quadratic Formula

Before we dive into solving our specific equation, let's first understand the quadratic formula. The quadratic formula is derived from the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we are trying to solve for. The quadratic formula itself is expressed as:

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

This formula tells us that the solutions for $x$ can be found by plugging in the values of $a$, $b$, and $c$ from our quadratic equation. The "$\pm$" symbol indicates that there are typically two solutions: one where we add the square root term and one where we subtract it. These solutions can be real numbers, imaginary numbers, or a combination of both. The part under the square root, $b^2 - 4ac$, is called the discriminant. The discriminant is particularly important because it tells us about the nature of the roots:

• If $b^2 - 4ac > 0$, the equation has two distinct real roots.
• If $b^2 - 4ac = 0$, the equation has one real root (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex roots.

The quadratic formula is not just a mathematical trick; it’s a fundamental tool for solving equations that model a variety of real-world phenomena. From physics to engineering to economics, quadratic equations pop up everywhere. Knowing how to use the quadratic formula efficiently and accurately is a valuable skill that will serve you well in numerous applications. Moreover, understanding the discriminant and what it tells us about the nature of the roots provides deeper insight into the solutions we obtain. Now, let's move on to applying this knowledge to our specific problem and see the formula in action.

Rewriting the Equation in Standard Form

Our given equation is $x^2 + 20 = 2x$. To use the quadratic formula, the first crucial step is to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves rearranging the terms so that all terms are on one side of the equation, and the equation is set equal to zero. In our case, we need to subtract $2x$ from both sides of the equation to achieve the standard form.

Starting with $x^2 + 20 = 2x$, we subtract $2x$ from both sides:

$x^2 + 20 - 2x = 2x - 2x$

This simplifies to:

$x^2 - 2x + 20 = 0$

Now, our equation is in the standard form $ax^2 + bx + c = 0$. This form is essential because it allows us to easily identify the coefficients $a$, $b$, and $c$, which are the key components we need to plug into the quadratic formula. In our rearranged equation, we can see that:

• $a = 1$ (the coefficient of $x^2$)
• $b = -2$ (the coefficient of $x$)
• $c = 20$ (the constant term)

Identifying these coefficients correctly is a critical step in solving the quadratic equation. Any mistake in this step will lead to incorrect values being plugged into the quadratic formula, and consequently, incorrect solutions. The process of rewriting the equation in standard form is not just about following a mathematical rule; it’s about setting the stage for accurate application of the quadratic formula. Once we have the equation in standard form and have correctly identified $a$, $b$, and $c$, we are ready to substitute these values into the quadratic formula and find the solutions for $x$. This careful preparation ensures that we are on the right track to solving the problem.

Applying the Quadratic Formula

Now that we have rewritten our equation $x^2 - 2x + 20 = 0$ in standard form and identified the coefficients $a = 1$, $b = -2$, and $c = 20$, we are ready to apply the quadratic formula:

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

We substitute the values of $a$, $b$, and $c$ into the formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(20)}}{2(1)}$$

Let's simplify this step by step. First, we deal with the terms outside the square root:

$$x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(20)}}{2}$$

Next, we simplify the expression under the square root:

$$x = \frac{2 \pm \sqrt{4 - 80}}{2}$$

$$x = \frac{2 \pm \sqrt{-76}}{2}$$

At this point, we notice that the value under the square root is negative, which means we will have complex solutions. This is because the discriminant ($b^2 - 4ac$) is negative. We can rewrite $\sqrt{-76}$ using imaginary numbers. Recall that $\sqrt{-1} = i$, where $i$ is the imaginary unit. We can factor out -1 from -76 and simplify the square root:

$$\sqrt{-76} = \sqrt{-1 \cdot 76} = \sqrt{-1} \cdot \sqrt{76} = i\sqrt{76}$$

Now, we can further simplify $\sqrt{76}$ by finding its prime factorization. $76 = 2 \cdot 2 \cdot 19 = 2^2 \cdot 19$, so $\sqrt{76} = \sqrt{2^2 \cdot 19} = 2\sqrt{19}$.

Therefore, $\sqrt{-76} = 2i\sqrt{19}$. Now we substitute this back into our equation:

$$x = \frac{2 \pm 2i\sqrt{19}}{2}$$

Finally, we simplify by dividing both terms in the numerator by 2:

$$x = 1 \pm i\sqrt{19}$$

Thus, the solutions for $x$ are $1 + i\sqrt{19}$ and $1 - i\sqrt{19}$. Applying the quadratic formula in this manner allows us to systematically solve for the roots of the equation, even when they are complex numbers.

Identifying the Correct Answer

After applying the quadratic formula and simplifying, we found the solutions for $x$ to be $1 + i\sqrt{19}$ and $1 - i\sqrt{19}$. These are complex roots, as indicated by the imaginary unit $i$. Now, we need to match our solutions with the given options to identify the correct answer.

Our solutions are in the form $1 \pm i\sqrt{19}$, which means we are looking for an option that matches this format. Let's review the given options:

A. $1 \pm \sqrt{21} i$ B. $-1 \pm \sqrt{19} i$ C. $1 \pm 2 \sqrt{19} i$ D. $1 \pm \sqrt{19} i$

Comparing our solutions with the options, we can see that option D, $1 \pm \sqrt{19} i$, matches our result exactly. Option A has a different value under the square root ($\sqrt{21}$ instead of $\sqrt{19}$), Option B has the wrong real part ($-1$ instead of $1$), and Option C has an extra factor of 2 multiplying the square root term ($2\sqrt{19}$ instead of $\sqrt{19}$). Therefore, option D is the correct answer.

The process of identifying the correct answer involves carefully comparing the solutions we obtained from the quadratic formula with the provided options. This step highlights the importance of accurate calculations and simplifications. It also demonstrates the need to pay close attention to detail when matching solutions, ensuring that we select the option that precisely matches our results.

Conclusion

In this article, we have demonstrated how to use the quadratic formula to solve the quadratic equation $x^2 + 20 = 2x$. We began by understanding the quadratic formula and its components, then rewrote the given equation in standard form to identify the coefficients $a$, $b$, and $c$. We carefully substituted these values into the quadratic formula, simplified the expression, and arrived at the complex solutions $x = 1 \pm i\sqrt{19}$. Finally, we matched our solutions with the provided options and identified the correct answer.

The quadratic formula is a versatile and essential tool for solving quadratic equations, especially when factoring is not straightforward. It is applicable to all quadratic equations, regardless of whether the roots are real or complex. Mastering the use of the quadratic formula not only enhances your problem-solving skills but also deepens your understanding of quadratic equations and their applications in various fields. Understanding the nature of the roots through the discriminant is also crucial, as it provides valuable insights into the solutions we obtain. This step-by-step guide should provide a solid foundation for tackling similar problems and further exploring the fascinating world of algebra.