Real Solutions Discriminant Quadratic Equation X^2 - 8x + 16 = 0

Delving into the world of quadratic equations, a fundamental concept in algebra, often involves determining the nature and number of solutions. The discriminant, a powerful tool derived from the quadratic formula, provides a straightforward method for unraveling this mystery. In this article, we will use the discriminant to analyze the quadratic equation $x^2 - 8x + 16 = 0$ and determine the number of real solutions it possesses.

Understanding the Discriminant

At the heart of quadratic equations lies the quadratic formula, which provides a general solution for equations in the form $ax^2 + bx + c = 0$, where a, b, and c are coefficients. The formula is expressed as:

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

The discriminant, denoted as Δ (Delta), is the expression under the square root sign in the quadratic formula: $b^2 - 4ac$. This simple yet insightful value reveals crucial information about the nature of the solutions to the quadratic equation. It allows us to determine whether the equation has two distinct real solutions, one real solution (a repeated root), or no real solutions (complex solutions).

The Discriminant's Significance

The discriminant acts as a key to unlock the secrets of a quadratic equation's solutions. Here's how it works:

• If Δ > 0: The equation has two distinct real solutions. This means the parabola represented by the quadratic equation intersects the x-axis at two different points.
• If Δ = 0: The equation has one real solution (a repeated root). The parabola touches the x-axis at exactly one point, representing a repeated root.
• If Δ < 0: The equation has no real solutions. The solutions are complex numbers, and the parabola does not intersect the x-axis.

Understanding these relationships is crucial for quickly assessing the nature of solutions without having to fully solve the quadratic equation.

Applying the Discriminant to $x^2 - 8x + 16 = 0$

Now, let's apply the discriminant to the specific quadratic equation $x^2 - 8x + 16 = 0$. To do this, we first identify the coefficients a, b, and c:

• a = 1 (the coefficient of $x^2$)
• b = -8 (the coefficient of x)
• c = 16 (the constant term)

Next, we substitute these values into the discriminant formula:

$$Δ = b^2 - 4ac = (-8)^2 - 4(1)(16) = 64 - 64 = 0$$

The discriminant, Δ, equals 0. According to our understanding of the discriminant's significance, this indicates that the quadratic equation has one real solution (a repeated root).

Interpreting the Result

The fact that the discriminant is zero tells us that the parabola represented by the equation $x^2 - 8x + 16 = 0$ touches the x-axis at exactly one point. This point represents the single real solution or the repeated root of the equation. To find this solution, we can either use the quadratic formula or recognize that the given quadratic expression is a perfect square trinomial.

In this case, $x^2 - 8x + 16$ can be factored as $(x - 4)^2$. Setting this equal to zero, we get $(x - 4)^2 = 0$, which implies x - 4 = 0, and thus x = 4. Therefore, the equation has one real solution, which is x = 4. This confirms our conclusion based on the discriminant.

Solving the Quadratic Equation

While the discriminant effectively reveals the number of real solutions, determining the actual solution(s) requires solving the quadratic equation. As mentioned earlier, recognizing patterns and applying appropriate methods can simplify the process. For the equation $x^2 - 8x + 16 = 0$, we can solve it through factoring:

1. Recognize the perfect square trinomial: The expression $x^2 - 8x + 16$ is a perfect square trinomial because it fits the pattern $a^2 - 2ab + b^2 = (a - b)^2$.
2. Factor the equation: Factoring $x^2 - 8x + 16$, we get $(x - 4)(x - 4) = 0$ or $(x - 4)^2 = 0$.
3. Solve for x: Taking the square root of both sides, we have $x - 4 = 0$.
4. Isolate x: Adding 4 to both sides, we find $x = 4$.

As we determined using the discriminant, the equation has one real solution, x = 4. This method reinforces the discriminant's accuracy in predicting the nature of solutions.

Alternative Solution Method: Using the Quadratic Formula

Although factoring provided a straightforward solution in this case, the quadratic formula serves as a universal tool for solving any quadratic equation. Let's apply it to $x^2 - 8x + 16 = 0$ to reinforce our understanding:

1. Identify coefficients: As before, a = 1, b = -8, and c = 16.

2. Apply the quadratic formula: Substitute the values into the formula:

   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(16)}}{2(1)}$$

3. Simplify: Simplify the expression:

   $$x = \frac{8 \pm \sqrt{64 - 64}}{2} = \frac{8 \pm \sqrt{0}}{2} = \frac{8 \pm 0}{2}$$

4. Solve for x: Since the square root of 0 is 0, we have:

   $$x = \frac{8}{2} = 4$$

The quadratic formula confirms that the equation has one real solution, x = 4. This consistency between methods underscores the reliability of both the discriminant and the quadratic formula in analyzing quadratic equations.

Conclusion

In conclusion, by applying the discriminant ($Δ = b^2 - 4ac$) to the quadratic equation $x^2 - 8x + 16 = 0$, we determined that it has one real solution. The discriminant value of 0 indicated a repeated root, which we confirmed by solving the equation through factoring and using the quadratic formula. The solution is x = 4.

Understanding the discriminant is essential for efficiently analyzing quadratic equations. It empowers us to quickly determine the nature and number of solutions, guiding us towards the appropriate method for solving the equation. Whether through factoring, using the quadratic formula, or other techniques, the discriminant provides a valuable foundation for tackling quadratic equations with confidence.

Therefore, the correct answer to the question "Using the discriminant, how many real solutions does the following quadratic equation have? $x^2 - 8x + 16 = 0$" is A. One real solution.

This exploration highlights the power and elegance of mathematical tools like the discriminant in unraveling the properties of equations. By mastering these concepts, we strengthen our problem-solving skills and gain a deeper appreciation for the beauty of mathematics.