Solving Quadratic Equations Expression Analysis And Solution

In the realm of mathematics, quadratic equations hold a position of immense importance. These equations, characterized by their second-degree polynomial nature, frequently arise in various scientific and engineering disciplines, demanding a deep understanding of their solutions. Among the tools for solving quadratic equations, the quadratic formula stands out as a cornerstone, providing a systematic approach to determine the roots of any quadratic equation.

The Quadratic Formula A Gateway to Solutions

The quadratic formula is a powerful tool that provides a direct method for finding the solutions, also known as roots, of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x represents the variable we aim to solve for. The quadratic formula provides a way to find the values of x that satisfy this equation. The formula is given by:

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

This formula reveals that a quadratic equation can have two solutions, which can be real or complex numbers, depending on the value of the discriminant ($b^2 - 4ac$). The discriminant plays a crucial role in determining the nature of the roots.

• If the discriminant ($b^2 - 4ac$) is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

Deciphering the Expression $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$

Let's analyze the given expression:

$\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$

This expression closely resembles the quadratic formula, and our goal is to identify the quadratic equation that can be solved using this expression. By carefully comparing this expression to the general form of the quadratic formula, we can extract the coefficients a, b, and c that correspond to the original quadratic equation.

Comparing the given expression with the quadratic formula:

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

we can make the following observations:

• -b in the formula corresponds to -3 in the expression, suggesting that b = 3.
• 2a in the formula corresponds to 2(10) in the expression, implying that a = 10.
• The term under the square root, $b^2 - 4ac$, corresponds to $(3)^2 + 4(10)(2)$ in the expression. Since we already know b = 3 and a = 10, we can deduce the value of c. The expression inside the square root can be rewritten as $3^2 - 4(10)(-2)$, which means -4ac is equivalent to 4(10)(2). Therefore, c = -2.

Now that we have identified the coefficients a = 10, b = 3, and c = -2, we can construct the corresponding quadratic equation:

$10x^2 + 3x - 2 = 0$

This equation represents the quadratic equation that can be solved using the given expression. By substituting the values of a, b, and c into the quadratic formula, we can obtain the solutions for x, which will be the same as those obtained from the original expression.

Identifying the Correct Equation

Now, let's examine the given options and determine which one matches the quadratic equation we derived:

A. $10x^2 = 3x + 2$ B. $2 = 3x + 10x^2$ C. $3x = 10x^2 - 2$ D. $10x^2 + 2 = -3x$

To make a direct comparison, we need to rewrite each equation in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$.

• Option A: $10x^2 = 3x + 2$ can be rewritten as $10x^2 - 3x - 2 = 0$.
• Option B: $2 = 3x + 10x^2$ can be rewritten as $10x^2 + 3x - 2 = 0$.
• Option C: $3x = 10x^2 - 2$ can be rewritten as $10x^2 - 3x - 2 = 0$.
• Option D: $10x^2 + 2 = -3x$ can be rewritten as $10x^2 + 3x + 2 = 0$.

Comparing these rewritten equations with the quadratic equation we derived, $10x^2 + 3x - 2 = 0$, we can see that Option B matches perfectly.

Therefore, the equation that can be solved using the expression $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$ for x is:

B. $2 = 3x + 10x^2$

Why This Expression? Exploring the Roots

The given expression is derived directly from the quadratic formula, which is used to find the roots of a quadratic equation. The roots are the values of x that make the quadratic equation equal to zero. In this case, the expression provides the two possible roots of the quadratic equation $10x^2 + 3x - 2 = 0$.

The expression incorporates the discriminant, which is the part under the square root, $b^2 - 4ac$. The discriminant determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. In this specific expression, the discriminant is $(3)^2 + 4(10)(2) = 9 + 80 = 89$, which is positive, indicating that the quadratic equation has two distinct real roots.

The $\pm$ symbol in the expression indicates that there are two possible solutions: one where we add the square root term and one where we subtract it. These two solutions represent the two points where the parabola defined by the quadratic equation intersects the x-axis.

Conclusion Navigating Quadratic Equations with Confidence

In summary, by carefully analyzing the given expression and comparing it with the quadratic formula, we were able to identify the corresponding quadratic equation. Option B, $2 = 3x + 10x^2$, is the equation that can be solved using the expression $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$. This exercise demonstrates the power of the quadratic formula in solving quadratic equations and highlights the importance of understanding its components.

Mastering quadratic equations and their solutions is crucial for success in various mathematical and scientific endeavors. The quadratic formula provides a reliable and efficient method for finding the roots of any quadratic equation, empowering us to tackle a wide range of problems with confidence. By understanding the relationship between the coefficients of a quadratic equation and the roots obtained from the quadratic formula, we gain a deeper appreciation for the elegance and power of mathematics.

Therefore, in conclusion, the ability to solve quadratic equations is a fundamental skill that opens doors to further exploration in mathematics and its applications. The quadratic formula serves as a key tool in this journey, enabling us to unlock the solutions to a vast array of problems and deepen our understanding of the mathematical world.