Discriminant And Real Solutions Of $-3=x^2+4x+1$

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In mathematics, particularly in algebra, the discriminant is a crucial concept when dealing with quadratic equations. Understanding the discriminant allows us to determine the nature and number of solutions a quadratic equation possesses without actually solving the equation. This article delves into the process of calculating the discriminant for a given quadratic equation and how to interpret its value to find the number of real number solutions. We will explore the quadratic formula, the standard form of a quadratic equation, and the significance of the discriminant in determining whether the solutions are real and distinct, real and equal, or complex.

The quadratic equation provided is $-3=x^2+4x+1$. Our primary goal is to determine the discriminant for this equation and, based on its value, ascertain how many real number solutions the equation has. This involves several key steps, including rewriting the equation in the standard quadratic form, identifying the coefficients, applying the discriminant formula, and interpreting the result. This comprehensive approach will not only provide the answer but also enhance the understanding of quadratic equations and their solutions.

Rewriting the Equation in Standard Quadratic Form

The first step in determining the discriminant is to rewrite the given equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. The given equation is $-3 = x^2 + 4x + 1$. To rewrite it in standard form, we need to move all terms to one side of the equation, setting the other side to zero. We can achieve this by adding 3 to both sides of the equation:

βˆ’3+3=x2+4x+1+3-3 + 3 = x^2 + 4x + 1 + 3

This simplifies to:

0=x2+4x+40 = x^2 + 4x + 4

Now, the equation is in the standard quadratic form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients. In this case:

  • a=1a = 1

  • b=4b = 4

  • c=4c = 4

Rewriting the equation in this standard form is essential because it allows us to easily identify the coefficients, which are crucial for calculating the discriminant. The coefficients $a$, $b$, and $c$ play a significant role in the quadratic formula and the discriminant, which we will discuss in the following sections. This foundational step ensures that we can accurately apply the discriminant formula and interpret the results correctly. Understanding the standard form of a quadratic equation is a fundamental concept in algebra, enabling us to analyze and solve various quadratic problems efficiently.

Calculating the Discriminant

Now that we have the quadratic equation in the standard form $x^2 + 4x + 4 = 0$, we can proceed to calculate the discriminant. The discriminant, often denoted by the Greek letter delta ($\Delta$), is a part of the quadratic formula that helps determine the nature of the roots of the quadratic equation. The formula for the discriminant is:

Ξ”=b2βˆ’4ac\Delta = b^2 - 4ac

Where:

  • a$ is the coefficient of the $x^2$ term.

  • b$ is the coefficient of the $x$ term.

  • c$ is the constant term.

In our equation, we have already identified the coefficients as:

  • a=1a = 1

  • b=4b = 4

  • c=4c = 4

Substituting these values into the discriminant formula, we get:

Ξ”=(4)2βˆ’4(1)(4)\Delta = (4)^2 - 4(1)(4)

Ξ”=16βˆ’16\Delta = 16 - 16

Ξ”=0\Delta = 0

Therefore, the discriminant of the quadratic equation $x^2 + 4x + 4 = 0$ is 0. The discriminant's value is critical because it provides direct insight into the number and type of solutions the quadratic equation has. A discriminant of 0 indicates that the quadratic equation has exactly one real solution (or a repeated real root). This is a significant finding, as it tells us that the parabola represented by the quadratic equation touches the x-axis at only one point. The calculation of the discriminant is a straightforward process, but its implications are profound in the context of solving quadratic equations.

Interpreting the Discriminant Value

After calculating the discriminant, the next crucial step is to interpret its value. The discriminant, denoted as $\Delta$, provides valuable information about the nature and number of solutions of a quadratic equation. The discriminant can be one of three possibilities: positive, zero, or negative, each indicating a different type of solution.

  1. If $\Delta > 0$ (Discriminant is positive): The quadratic equation has two distinct real solutions. This means that the parabola represented by the quadratic equation intersects the x-axis at two different points. The solutions are real numbers and are not equal.
  2. If $\Delta = 0$ (Discriminant is zero): The quadratic equation has exactly one real solution, which is sometimes referred to as a repeated root or a double root. This means that the parabola touches the x-axis at only one point. The single solution can be found using the quadratic formula, and it is a real number.
  3. If $\Delta < 0$ (Discriminant is negative): The quadratic equation has no real solutions. Instead, it has two complex solutions. This means that the parabola does not intersect the x-axis at any point. The solutions involve imaginary numbers and are complex conjugates.

In our case, we calculated the discriminant to be:

Ξ”=0\Delta = 0

This means that the quadratic equation $x^2 + 4x + 4 = 0$ has exactly one real solution. The parabola touches the x-axis at one point, indicating a repeated root. Interpreting the discriminant is a critical skill in algebra, as it allows us to quickly determine the nature of the solutions without fully solving the quadratic equation. This interpretation guides us in understanding the behavior of the quadratic function and its graphical representation.

Determining the Number of Real Solutions

Based on the interpretation of the discriminant value, we can now determine the number of real solutions for the given quadratic equation $-3 = x^2 + 4x + 1$. We have already established that the equation in standard form is $x^2 + 4x + 4 = 0$, and the calculated discriminant is $\Delta = 0$.

As discussed in the previous section, a discriminant of 0 indicates that the quadratic equation has exactly one real solution. This means there is one value of $x$ that satisfies the equation. Graphically, this corresponds to the parabola touching the x-axis at a single point, which is the vertex of the parabola.

Therefore, the quadratic equation $-3 = x^2 + 4x + 1$ has one real solution. This single solution can be found by either factoring the quadratic equation, completing the square, or using the quadratic formula. Since the discriminant is 0, the quadratic formula will yield a single real root.

The number of real solutions is a key characteristic of a quadratic equation, and the discriminant provides a straightforward method to determine this without needing to solve the equation explicitly. Understanding this connection between the discriminant and the number of real solutions is fundamental in solving and analyzing quadratic equations.

Solving the Quadratic Equation

To further illustrate the concept and verify our findings, let's solve the quadratic equation $x^2 + 4x + 4 = 0$. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, the equation is easily factorable.

The equation $x^2 + 4x + 4 = 0$ can be factored as:

(x+2)(x+2)=0(x + 2)(x + 2) = 0

Or equivalently:

(x+2)2=0(x + 2)^2 = 0

To find the solution, we set each factor equal to zero:

x+2=0x + 2 = 0

Solving for $x$, we get:

x=βˆ’2x = -2

Since both factors are the same, we have a repeated root. This confirms our earlier conclusion based on the discriminant value, which indicated that there is exactly one real solution. The solution to the quadratic equation is $x = -2$.

Alternatively, we could have used the quadratic formula to solve the equation. The quadratic formula is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Plugging in the values $a = 1$, $b = 4$, and $c = 4$, we get:

x=βˆ’4Β±42βˆ’4(1)(4)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}

x=βˆ’4Β±16βˆ’162x = \frac{-4 \pm \sqrt{16 - 16}}{2}

x=βˆ’4Β±02x = \frac{-4 \pm \sqrt{0}}{2}

x=βˆ’4Β±02x = \frac{-4 \pm 0}{2}

x=βˆ’42x = \frac{-4}{2}

x=βˆ’2x = -2

Again, we find that the equation has one real solution, $x = -2$, which further validates our analysis using the discriminant.

Conclusion

In summary, we have successfully determined the discriminant for the quadratic equation $-3 = x^2 + 4x + 1$ and, based on its value, identified the number of real solutions. By rewriting the equation in the standard quadratic form $x^2 + 4x + 4 = 0$, we identified the coefficients $a = 1$, $b = 4$, and $c = 4$. We then calculated the discriminant using the formula $\Delta = b^2 - 4ac$, which yielded a value of 0.

The discriminant value of 0 indicates that the quadratic equation has exactly one real solution. This means the parabola represented by the equation touches the x-axis at a single point. We verified this by solving the equation through factoring and using the quadratic formula, both methods yielding the single solution $x = -2$.

Understanding the discriminant is crucial in analyzing quadratic equations, as it provides a quick and effective way to determine the nature and number of solutions without explicitly solving the equation. The discriminant's value helps us classify the solutions as two distinct real solutions (if $\Delta > 0$), one real solution (if $\Delta = 0$), or two complex solutions (if $\Delta < 0$). This knowledge is fundamental in various applications of quadratic equations in mathematics and other fields.

This comprehensive analysis demonstrates the importance of the discriminant in understanding the behavior and solutions of quadratic equations. By mastering these concepts, one can effectively tackle a wide range of algebraic problems and applications involving quadratic functions.