Present A Quadratic Equation In The Form Ax²
Present A Quadratic Equation In The Form Ax2
Present a quadratic equation in the form ax^2 + bx + c = 0 where a > 1. How many solutions does your quadratic have based on the discriminant? Pick TWO ways to find the specific solutions or show that there is no solution: Quadratic Formula, Graphing, Factoring, Square Root Property, Completing the Square. Why did you choose those two specific methods versus the others? Make sure that you do NOT use the same quadratic equation presented by one of your peers. Writing Requirements: Minimum of 2 posts (1 initial & 1 follow-up) with first post expected by Wednesday. APA format for in-text citations and list of references.
Paper For Above instruction
The quadratic equation I have selected for this assignment is 3x^2 - 12x + 7 = 0, where the coefficient a = 3, which satisfies the condition that a > 1. This quadratic equation will serve as the basis for analyzing the number of solutions it has, as well as demonstrating two methods for finding these solutions.
To determine the number of solutions, I will first evaluate the discriminant, Δ, which is computed as Δ = b^2 - 4ac. For the given quadratic, the coefficients are a = 3, b = -12, and c = 7. Substituting these into the discriminant formula yields:
Δ = (-12)^2 - 4(3)(7) = 144 - 84 = 60.
Since the discriminant is positive (Δ = 60 > 0), the quadratic has two distinct real solutions. This indicates that the parabola represented by the quadratic crosses the x-axis at two points.
For finding the specific solutions, I will employ the quadratic formula and graphing methods. These are chosen because the quadratic formula provides an exact algebraic solution regardless of the coefficients, and graphing offers a visual confirmation of the nature and approximate location of the roots.
The quadratic formula is given by:
x = [-b ± √Δ] / (2a).
Applying the values:
x = [12 ± √60] / (6).
Simplifying further:
x = [12 ± √(4 * 15)] / (6) = [12 ± 2√15] / (6).
Dividing numerator and denominator by 2:
x = [6 ± √15] / (3).
Therefore, the solutions are:
x = (6 + √15) / 3 and x = (6 - √15) / 3.
Next, graphing allows us to visually verify these solutions. Plotting the quadratic function y = 3x^2 - 12x + 7, the parabola intersects the x-axis at two points corresponding to the solutions obtained algebraically. Using graphing tools like Desmos or graphing calculators, we observe the parabola crossing the x-axis near x ≈ 2.28 and x ≈ 0.39, which aligns closely with the algebraic solutions.
I chose these two methods because the quadratic formula provides an exact, reliable solution, especially when the discriminant is positive, and the equations involve irrational numbers. Graphing complements this by offering an intuitive understanding of where the roots lie and confirming the number of solutions visually. Other methods such as factoring are less suitable here because the quadratic does not easily factor into rational roots, and completing the square might be more cumbersome due to the coefficients involved. The square root property is primarily useful for equations in a specific form, but for quadratics with mixed coefficients, the quadratic formula is more straightforward.
In conclusion, for the quadratic equation 3x^2 - 12x + 7 = 0, the discriminant indicates two real solutions. The quadratic formula provides precise solutions as (6 + √15)/3 and (6 - √15)/3, while graphing visually confirms the approximate x-intercepts at about 2.28 and 0.39. The combination of these methods yields a comprehensive understanding of the solutions' nature and location.
References
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