Mastering The Quadratic Formula: Solving $5x^2 + 3x - 4 = 0$

Understanding how to use the quadratic formula is a fundamental skill in algebra. It's a powerful tool that allows us to find the solutions (or roots) of any quadratic equation, which is an equation of the form $ax^2 + bx + c = 0$. Today, we're going to dive deep into correctly applying this formula to a specific equation: $5x^2 + 3x - 4 = 0$. We'll break down the formula, identify the coefficients, and walk through the steps to ensure you can confidently solve similar problems. Many students find quadratic equations a bit daunting at first, but with a clear explanation and a bit of practice, you'll be solving them like a pro. The beauty of the quadratic formula lies in its universality; it works for all quadratic equations, regardless of whether they can be factored easily or not. This makes it an indispensable part of your mathematical toolkit.

Understanding the Quadratic Formula and Its Components

The quadratic formula is derived from the general form of a quadratic equation, $ax^2 + bx + c = 0$. The formula itself is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. To use it correctly, we need to identify the values of $a$, $b$, and $c$ from our specific equation. In the equation $5x^2 + 3x - 4 = 0$, we can clearly see that: $a = 5$ (the coefficient of the $x^2$ term), $b = 3$ (the coefficient of the $x$ term), and $c = -4$ (the constant term). It's crucial to pay close attention to the signs of these coefficients. A common mistake is to forget the negative sign for $c$, which can lead to an incorrect final answer. The $\pm$ symbol in the formula indicates that there are typically two possible solutions for $x$: one where we add the square root term and one where we subtract it. These two solutions arise because a quadratic equation represents a parabola, which can intersect the x-axis at zero, one, or two points. The term inside the square root, $b^2 - 4ac$, is called the discriminant, and it tells us about the nature of the roots (real and distinct, real and equal, or complex). Getting these values right is the first and most important step in correctly applying the formula. If you misidentify any of these coefficients, the entire calculation will be off. So, always double-check your $a$, $b$, and $c$ values before plugging them into the formula.

Applying the Formula to $5x^2 + 3x - 4 = 0$

Now that we have identified $a = 5$, $b = 3$, and $c = -4$, we can substitute these values into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Plugging in our values, we get: $x = \frac{-(3) \pm \sqrt{(3)^2 - 4(5)(-4)}}{2(5)}$. Let's carefully examine each part of this expression. The $-b$ term becomes $-3$. The $b^2$ term becomes $(3)^2$, which is $9$. The $-4ac$ term becomes $-4(5)(-4)$. When we multiply this out, $-4 \times 5$ is $-20$, and $-20 \times -4$ is $+80$. So, the expression inside the square root, the discriminant, is $9 + 80 = 89$. The denominator, $2a$, becomes $2(5)$, which is $10$. Therefore, the correct application of the quadratic formula to the equation $5x^2 + 3x - 4 = 0$ yields: $x = \frac{-3 \pm \sqrt{89}}{10}$. This detailed substitution process highlights why precision is key. Each number and sign must be handled correctly. The process is not just about memorizing the formula but understanding how to adapt it to different equations. This step-by-step substitution ensures that we are accurately representing the original equation within the structure of the formula. We're not just guessing; we're systematically replacing the variables $a$, $b$, and $c$ with their corresponding values from the given quadratic equation. This methodical approach minimizes errors and builds confidence in your mathematical abilities. The result is a precise expression ready for further simplification or calculation.

Evaluating the Options and Identifying the Correct Equation

We are given four options for the correct equation showing the quadratic formula applied to $5x^2 + 3x - 4 = 0$. Let's analyze each one based on our correct substitution: $x = \frac{-3 \pm \sqrt{(3)^2 - 4(5)(-4)}}{2(5)}$.

• Option A: $x=\frac{-3 \pm \sqrt{(3)^2-4(5)(-4)}}{2(5)}$. This option matches our derived equation exactly. The $-b$ term is $-3$, $b^2$ is $(3)^2$, $-4ac$ is $-4(5)(-4)$, and $2a$ is $2(5)$. All components are correctly substituted.

• Option B: $x=\frac{3 \pm \sqrt{(3)^2+4(5)(-4)}}{2(5)}$. This option has two errors. The $-b$ term should be $-3$, not $3$. Additionally, the $-4ac$ term should be $+80$ (since $-4 \times 5 \times -4 = +80$), but here it is written as $+4(5)(-4)$, which results in $-80$. This incorrect sign within the discriminant is a critical error.

• Option C: $x=\frac{3 \pm \sqrt{(3)^2-4(5)(-4)}}{2(5)}$. This option also has an error in the $-b$ term, which should be $-3$. The rest of the formula appears correct, but the initial sign error makes it incorrect.

• Option D: $x=\frac{-3}{}$ (incomplete option, but based on the beginning, it seems to be missing the \pm \sqrt{b^2-4ac}}{2a} part and likely contains other errors if fully written).

Based on this thorough comparison, Option A is the only equation that correctly shows the substitution of the coefficients $a=5$, $b=3$, and $c=-4$ into the quadratic formula. It accurately reflects the $-b$ term, the $b^2$ term, the $-4ac$ term (including the signs resulting from the multiplication), and the $2a$ term. Getting to this point requires careful attention to detail, especially with signs and the order of operations within the formula. It's not uncommon for students to make small errors here, so reviewing each part of the formula and comparing it to the options is a vital step in problem-solving. This meticulous verification process ensures that the foundation for solving the equation is solid. Without the correct setup using the quadratic formula, any subsequent calculations would be invalid.

The Importance of Correct Substitution and Sign Conventions

The journey to mastering the quadratic formula truly hinges on correct substitution and unwavering attention to sign conventions. Let's reiterate why this is so crucial, especially when dealing with equations like $5x^2 + 3x - 4 = 0$. The formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, acts as a universal key, but the lock it opens is sensitive to the exact values of $a$, $b$, and $c$. In our equation, $a=5$, $b=3$, and $c=-4$. Notice that $c$ is negative. This is a common area where errors creep in. When we substitute $c=-4$ into the $-4ac$ part of the formula, we get $-4(5)(-4)$. The multiplication of three numbers, two of which are negative, results in a positive number: $-4 \times 5 = -20$, and $-20 \times -4 = +80$. If one incorrectly substitutes $c=4$ or makes a mistake in the multiplication of signs, the discriminant ($b^2 - 4ac$) would be wrong. For instance, if $c$ was mistakenly taken as positive 4, then $-4ac$ would become $-4(5)(4) = -80$. The discriminant would then be $3^2 - 80 = 9 - 80 = -71$. This would lead to taking the square root of a negative number, which, unless you're working with complex numbers, would indicate an error in your setup or that the equation has no real solutions. The term $-b$ also requires care. Since $b=3$, $-b$ is simply $-3$. If $b$ had been negative, say $b=-3$, then $-b$ would be $-(-3) = +3$. Forgetting this double negative can flip the sign of the entire numerator. The denominator, $2a$, is generally more straightforward, but still requires accurate multiplication: $2 \times 5 = 10$. Each of these steps builds upon the last. A small slip-up early on can cascade into a completely incorrect final answer. Therefore, practicing methodical substitution, perhaps by writing out each substitution step explicitly as we did in analyzing the options, can significantly reduce errors. It reinforces the correct placement of each coefficient and the application of the formula's operations. This meticulousness is what separates a correct solution from an incorrect one when using the quadratic formula. It's a testament to the idea that in mathematics, attention to detail isn't just helpful; it's essential for accuracy and understanding.

Conclusion: The Power of Precision with the Quadratic Formula

We've thoroughly dissected the quadratic formula and its application to the equation $5x^2 + 3x - 4 = 0$. By carefully identifying $a=5$, $b=3$, and $c=-4$, and substituting these values into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we arrived at the correct expression: $x = \frac{-3 \pm \sqrt{(3)^2 - 4(5)(-4)}}{2(5)}$. This meticulous process, especially paying close attention to the signs of the coefficients and the operations within the formula, is paramount. As we saw, options B and C contained critical sign errors, and option D was incomplete. Option A stands out as the only one that accurately represents the correct application of the quadratic formula for this specific equation. Mastering the quadratic formula isn't just about memorizing a sequence of symbols; it's about understanding the structure of quadratic equations and applying a proven method with precision. This skill will serve you well in various areas of mathematics, from solving word problems to graphing parabolas and understanding the behavior of functions. Keep practicing, and don't be afraid to double-check your work. For further exploration into quadratic equations and their properties, you might find resources like Khan Academy incredibly helpful.