Standard Form & Quadratic Formula: Solve $3x^2 - 8x + 7 = 0$

When tackling quadratic equations, understanding their standard form is the crucial first step. The standard form of a quadratic equation is generally expressed as $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. This form is incredibly important because it provides a consistent framework for applying various solving methods, including the powerful quadratic formula. Our specific equation, $3x^2 - 8x + 7 = 0$, is already perfectly aligned with this standard form. Here, we can clearly identify our coefficients: $a = 3$, $b = -8$, and $c = 7$. Recognizing these values is the gateway to unlocking the solution using the quadratic formula. The beauty of the standard form lies in its universality; no matter how a quadratic equation is initially presented, it can almost always be rearranged into this tidy format, making it amenable to a host of algebraic techniques. For instance, if we had an equation like $5x = 2x^2 - 1$, we would first need to move all terms to one side to achieve standard form: $2x^2 - 5x - 1 = 0$. This reorganization is fundamental, ensuring that we are comparing apples to apples when using methods designed for this specific structure. The coefficients '$a$', '$b$', and '$c$' hold the key information about the parabola that the equation represents – its width, direction, and where it crosses the x-axis. Therefore, the initial step of ensuring an equation is in standard form is not merely a formality but a critical diagnostic and preparatory action for any subsequent analysis or solution-finding endeavor. It simplifies complex expressions into a predictable pattern, paving the way for systematic problem-solving.

Once our equation is in the coveted standard form, $ax^2 + bx + c = 0$, we can confidently deploy the quadratic formula to find the solutions (or roots) for 'x'. The formula itself is a remarkable piece of mathematical machinery, derived through a process called completing the square, and it guarantees a solution for any quadratic equation. The formula is presented as: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. To use it effectively, we simply substitute the values of 'a', 'b', and 'c' from our standard form equation into this formula. For our equation, $3x^2 - 8x + 7 = 0$, we have $a = 3$, $b = -8$, and $c = 7$. Let's meticulously plug these values into the quadratic formula: $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(7)}}{2(3)}$. Simplifying this expression step-by-step is essential to avoid errors. First, $-(-8)$ becomes $8$. Next, $(-8)^2$ equals $64$. Then, $4(3)(7)$ equals $84$. So, the expression under the square root, known as the discriminant, becomes $64 - 84$, which is $-20$. The denominator, $2(3)$, is $6$. Thus, our formula now reads: $x = \frac{8 \pm \sqrt{-20}}{6}$. The presence of a negative number under the square root indicates that the solutions will be complex numbers. We can simplify $\sqrt{-20}$ by recognizing that $\sqrt{-20} = \sqrt{20} \times \sqrt{-1}$. Since $\sqrt{-1}$ is represented by the imaginary unit '$i$', and $\sqrt{20}$ can be simplified to $\sqrt{4 \times 5} = 2\sqrt{5}$, we have $\sqrt{-20} = 2i\sqrt{5}$. Substituting this back into our equation for 'x', we get $x = \frac{8 \pm 2i\sqrt{5}}{6}$. To present the final solutions in their simplest form, we can divide both terms in the numerator and the denominator by their greatest common divisor, which is 2. This gives us the final solutions: $x = \frac{4 \pm i\sqrt{5}}{3}$. These are our two distinct complex roots for the given quadratic equation. The quadratic formula is an indispensable tool, providing a reliable pathway to solutions whether they are real or complex, rational or irrational.

Understanding the Discriminant

The expression $b^2 - 4ac$, found inside the square root of the quadratic formula, is known as the discriminant. Its value provides invaluable insight into the nature of the roots (solutions) of the quadratic equation without us having to calculate the roots themselves. This is a powerful concept in understanding quadratic behavior. Let's break down what the discriminant tells us:

• If $b^2 - 4ac > 0$: The discriminant is positive. This signifies that there are two distinct real roots. These roots will be rational if the discriminant is a perfect square, and irrational if it is not. Geometrically, this means the parabola represented by the quadratic equation intersects the x-axis at two different points.

• If $b^2 - 4ac = 0$: The discriminant is zero. This indicates that there is exactly one real root, often referred to as a repeated root or a double root. In this scenario, the vertex of the parabola lies directly on the x-axis, meaning the parabola touches the x-axis at a single point.

• If $b^2 - 4ac < 0$: The discriminant is negative. This is the case for our equation $3x^2 - 8x + 7 = 0$, where the discriminant was $-20$. A negative discriminant means there are two distinct complex conjugate roots. These roots involve the imaginary unit '$i$' and are not real numbers. Graphically, this implies that the parabola does not intersect the x-axis at all; it either lies entirely above or entirely below it.

In our specific problem, $3x^2 - 8x + 7 = 0$, we calculated the discriminant as $b^2 - 4ac = (-8)^2 - 4(3)(7) = 64 - 84 = -20$. Since $-20$ is less than zero, we correctly concluded that the equation has two distinct complex conjugate roots. This analysis of the discriminant is a vital part of understanding quadratic equations. It acts as a quick check, helping us anticipate the type of solutions we will find before we even complete the full calculation using the quadratic formula. It’s like a preview of the solution's nature, enhancing our comprehension of the underlying mathematical structure. Mastering the discriminant allows for a more nuanced and efficient approach to solving and interpreting quadratic equations, saving time and preventing potential misinterpretations of results. It is a fundamental concept that bridges the gap between algebraic manipulation and graphical representation, offering a deeper appreciation for the behavior of these essential functions.

Step-by-Step Solution Summary

Let's consolidate the process for solving the quadratic equation $3x^2 - 8x + 7 = 0$ using the quadratic formula. The journey begins with ensuring the equation is in standard form, which it already is: $ax^2 + bx + c = 0$. From this, we identify our coefficients: $a = 3$, $b = -8$, and $c = 7$. These values are the essential ingredients for the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step 1: Substitute the coefficients into the quadratic formula.

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(7)}}{2(3)}$

Step 2: Simplify the terms within the formula.

• $-(-8)$ becomes $8$.
• $(-8)^2$ becomes $64$.
• $4(3)(7)$ becomes $84$.
• $2(3)$ becomes $6$.

So the formula transforms into:

$x = \frac{8 \pm \sqrt{64 - 84}}{6}$

Step 3: Calculate the discriminant.

The discriminant is $64 - 84 = -20$. This tells us we will have complex roots.

$x = \frac{8 \pm \sqrt{-20}}{6}$

Step 4: Simplify the square root of the discriminant.

$\\sqrt{-20} = \sqrt{20} \times \sqrt{-1} = \sqrt{4 \times 5} \times i = 2\sqrt{5}i$

Substituting this back:

$x = \frac{8 \pm 2i\sqrt{5}}{6}$

Step 5: Simplify the entire expression.

Divide each term in the numerator and the denominator by their greatest common divisor, which is 2:

$x = \frac{8/2 \pm (2i\sqrt{5})/2}{6/2}$

$x = \frac{4 \pm i\sqrt{5}}{3}$

Therefore, the two solutions for the equation $3x^2 - 8x + 7 = 0$ are $x = \frac{4 + i\sqrt{5}}{3}$ and $x = \frac{4 - i\sqrt{5}}{3}$. This methodical approach ensures accuracy and provides a clear path to the solution, regardless of the complexity of the numbers involved.

In conclusion, the process of solving quadratic equations is made significantly more manageable and systematic by first placing them in standard form and then applying the quadratic formula. This formula, coupled with an understanding of the discriminant, provides a comprehensive toolkit for finding solutions, whether they manifest as real or complex numbers. The ability to transform an equation into $ax^2 + bx + c = 0$ and then use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ is fundamental in algebra and has wide-ranging applications in science, engineering, and economics. If you're looking for more resources on quadratic equations or algebra in general, you might find the Khan Academy website to be an excellent and free resource for learning and practice.