Solving Quadratic Equations Using The Quadratic Formula A Step By Step Guide

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In this comprehensive guide, we will delve into the process of identifying the quadratic equation that can be solved using the given expression: $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$. This expression is derived from the quadratic formula, a fundamental tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. We will meticulously analyze the given expression, dissect its components, and relate them back to the quadratic formula to pinpoint the correct quadratic equation from the provided options. This exploration will not only solidify your understanding of the quadratic formula but also enhance your ability to manipulate and solve quadratic equations efficiently.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions (also called roots or zeros) of any quadratic equation. A quadratic equation is an equation of the form:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are constants, and a is not equal to 0. The quadratic formula provides the values of x that satisfy this equation, and it is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant provides valuable information 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.

Deconstructing the Given Expression

Now, let's carefully examine the expression provided: $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$. Our goal is to match the components of this expression with the quadratic formula to determine the corresponding values of a, b, and c. By identifying these coefficients, we can reconstruct the original quadratic equation.

Comparing the given expression with the quadratic formula, we can make the following observations:

  • The term "-b" in the formula corresponds to "-3" in the expression. This suggests that b = 3.
  • The denominator "2a" corresponds to "2(10)", indicating that a = 10.
  • The expression under the square root, $b^2 - 4ac$, corresponds to $(3)^2 + 4(10)(2)$. Note the plus sign instead of a minus sign in the formula. This subtle difference is crucial and requires careful consideration. The expression can be rewritten as $3^2 - 4(10)(-2)$, which aligns with the discriminant form $b^2 - 4ac$. Thus, c = -2.

Reconstructing the Quadratic Equation

Having identified the coefficients a = 10, b = 3, and c = -2, we can now construct the quadratic equation in the standard form $ax^2 + bx + c = 0$. Substituting the values, we get:

10x2+3x−2=010x^2 + 3x - 2 = 0

This equation is the quadratic equation that can be solved using the given expression.

Analyzing the Answer Choices

Now, let's analyze the provided answer choices to identify the equation that matches our reconstructed equation:

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

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

  • Option A: $10x^2 = 3x + 2$ can be rewritten as $10x^2 - 3x - 2 = 0$. This does not match our reconstructed equation.
  • Option B: $2 = 3x + 10x^2$ can be rewritten as $10x^2 + 3x - 2 = 0$. This matches our reconstructed equation.
  • Option C: $3x = 10x^2 - 2$ can be rewritten as $10x^2 - 3x - 2 = 0$. This does not match our reconstructed equation.
  • Option D: $10x^2 + 2 = -3x$ can be rewritten as $10x^2 + 3x + 2 = 0$. This does not match our reconstructed equation.

Therefore, the correct answer is Option B, as it is the only equation that can be rearranged into the standard form $10x^2 + 3x - 2 = 0$, which is the equation we derived from the given expression.

Step-by-Step Solution

Let's summarize the step-by-step solution to solidify your understanding:

  1. Understand the Quadratic Formula: Begin by recalling the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which solves equations of the form $ax^2 + bx + c = 0$.
  2. Deconstruct the Given Expression: Carefully compare the given expression, $\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}$, to the quadratic formula. Identify the corresponding parts to deduce the values of a, b, and c.
  3. Identify 'b': The term "-b" in the formula corresponds to "-3" in the expression, so b = 3.
  4. Identify 'a': The denominator "2a" corresponds to "2(10)", indicating that a = 10.
  5. Identify 'c': The discriminant portion, $\sqrt{b^2 - 4ac}$, corresponds to $\sqrt{(3)^2+4(10)(2)}$. Rewrite the expression under the square root as $3^2 - 4(10)(-2)$, so c = -2.
  6. Reconstruct the Quadratic Equation: Substitute the values a = 10, b = 3, and c = -2 into the standard form $ax^2 + bx + c = 0$ to get $10x^2 + 3x - 2 = 0$.
  7. Analyze the Answer Choices: Rewrite each answer choice in the standard form $ax^2 + bx + c = 0$ and compare it with the reconstructed equation.
  8. Match the Equation: Identify the answer choice that matches the reconstructed equation $10x^2 + 3x - 2 = 0$. In this case, option B, $2 = 3x + 10x^2$, matches when rewritten as $10x^2 + 3x - 2 = 0$.

Conclusion

In conclusion, by meticulously dissecting the given expression and comparing its components with the quadratic formula, we successfully identified the quadratic equation that can be solved using the expression. The process involved understanding the structure of the quadratic formula, identifying the coefficients a, b, and c, reconstructing the quadratic equation, and comparing it with the provided options. This exercise highlights the importance of a strong understanding of the quadratic formula and the ability to manipulate algebraic expressions effectively. Mastering these skills will undoubtedly enhance your problem-solving capabilities in mathematics. The correct answer is Option B: $2=3 x+10 x^2$.

1. What is the quadratic formula, and when should I use it?

The quadratic formula is a formula used to find the solutions (roots) of a quadratic equation in the form $ax^2 + bx + c = 0$, where a, b, and c are constants and a ≠ 0. The formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

You should use the quadratic formula when you need to find the roots of a quadratic equation, especially when the equation is difficult to factor or when you need an exact solution (including irrational or complex roots). It's a versatile tool that works for all quadratic equations.

2. How do I identify the values of a, b, and c in a quadratic equation?

To identify the values of a, b, and c, you first need to rewrite the quadratic equation in the standard form, which is $ax^2 + bx + c = 0$. Once the equation is in this form:

  • a is the coefficient of the $x^2$ term.
  • b is the coefficient of the x term.
  • c is the constant term (the term without any x).

For example, in the equation $3x^2 - 5x + 2 = 0$, a = 3, b = -5, and c = 2.

3. What is the discriminant, and what does it tell me about the roots of a quadratic equation?

The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. The discriminant provides valuable information about the nature of the roots of the quadratic equation:

  • If $b^2 - 4ac > 0$, the equation has two distinct real roots. This means there are two different real numbers that satisfy the equation.
  • If $b^2 - 4ac = 0$, the equation has one real root (a repeated root). This means there is exactly one real number that satisfies the equation, and it occurs twice.
  • If $b^2 - 4ac < 0$, the equation has two complex roots. This means there are no real numbers that satisfy the equation; the roots are complex numbers (involving the imaginary unit i).

4. Can I use the quadratic formula for equations that are not in the standard form $ax^2 + bx + c = 0$?

Yes, you can use the quadratic formula for equations that are not initially in the standard form, but you must first rewrite the equation in the standard form. This typically involves rearranging terms so that all terms are on one side of the equation and the other side is zero. Once the equation is in the standard form, you can easily identify the values of a, b, and c and apply the quadratic formula.

5. Are there other methods to solve quadratic equations besides the quadratic formula?

Yes, there are several other methods to solve quadratic equations, including:

  • Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It's often the quickest method when the quadratic equation can be easily factored.
  • Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial on one side. It's a useful method for understanding the derivation of the quadratic formula and can be used to solve any quadratic equation.
  • Graphing: This method involves graphing the quadratic equation (a parabola) and finding the x-intercepts (the points where the graph crosses the x-axis), which represent the roots of the equation. This method is particularly useful for visualizing the roots and estimating their values.

The choice of method often depends on the specific equation and your personal preference. The quadratic formula is the most versatile method as it works for all quadratic equations, but factoring or completing the square may be quicker in some cases.

Quadratic Equation, Quadratic Formula, Solving Quadratic Equations, Roots of Quadratic Equations, Discriminant, Coefficients, Standard Form, Factoring, Completing the Square, Graphing, Algebra, Mathematics