Solving Quadratic Equations Find The Value Of X In X^2 = 6x - 9
In the realm of mathematics, quadratic equations hold a fundamental position, often encountered in various fields ranging from physics to engineering. These equations, characterized by their second-degree polynomial form, present a fascinating challenge to solve, revealing the hidden values of the unknown variable, typically denoted as 'x'.
Delving into the Quadratic Equation x^2 = 6x - 9
Our focus lies on the specific quadratic equation x^2 = 6x - 9. To embark on the journey of finding the value or values of 'x' that satisfy this equation, we must first transform it into its standard form. The standard form of a quadratic equation is expressed as:
ax^2 + bx + c = 0
where 'a', 'b', and 'c' represent constant coefficients. To achieve this standard form, we need to rearrange the terms of our equation, bringing all terms to one side and setting the equation equal to zero. Subtracting 6x and adding 9 to both sides of the equation, we arrive at:
x^2 - 6x + 9 = 0
Now, our equation stands in the familiar standard form, ready for us to unleash our problem-solving techniques.
Unveiling the Methods to Solve Quadratic Equations
Several powerful methods exist to solve quadratic equations, each with its own strengths and applicability. Let's explore three prominent approaches:
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Factoring: This method involves breaking down the quadratic expression into two linear factors. If we can successfully factor the expression, we can then set each factor equal to zero and solve for 'x'.
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Quadratic Formula: This versatile formula provides a direct solution for 'x', regardless of whether the equation can be factored or not. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where 'a', 'b', and 'c' are the coefficients of the quadratic equation in standard form.
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Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side, allowing us to easily solve for 'x' by taking the square root of both sides.
Applying Factoring to Our Equation
In our case, the quadratic expression x^2 - 6x + 9 lends itself beautifully to factoring. We seek two numbers that add up to -6 (the coefficient of the 'x' term) and multiply to 9 (the constant term). After some contemplation, we realize that -3 and -3 perfectly fit the bill. Therefore, we can factor the expression as:
(x - 3)(x - 3) = 0
Solving for x
Now, we have the factored form of our equation. To find the values of 'x' that satisfy the equation, we set each factor equal to zero:
x - 3 = 0
Adding 3 to both sides, we obtain:
x = 3
Notice that both factors are identical, leading to the same solution. This indicates that our quadratic equation has a repeated root.
The Solution: x = 3
Therefore, the value of 'x' that satisfies the quadratic equation x^2 = 6x - 9 is x = 3. This corresponds to option C in the given choices.
Verification
To ensure the accuracy of our solution, we can substitute x = 3 back into the original equation:
(3)^2 = 6(3) - 9
9 = 18 - 9
9 = 9
The equation holds true, confirming that our solution x = 3 is indeed correct.
Exploring Alternative Methods: Quadratic Formula and Completing the Square
While factoring provided a straightforward path to the solution in this case, let's briefly explore how the other methods, the quadratic formula and completing the square, would lead us to the same answer.
Applying the Quadratic Formula
Recall the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, x^2 - 6x + 9 = 0, we have a = 1, b = -6, and c = 9. Substituting these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)^2 - 4 * 1 * 9)) / (2 * 1)
x = (6 ± √(36 - 36)) / 2
x = (6 ± √0) / 2
x = 6 / 2
x = 3
As we can see, the quadratic formula yields the same solution, x = 3.
Completing the Square
To complete the square, we manipulate the equation to create a perfect square trinomial. Starting with:
x^2 - 6x + 9 = 0
Notice that the left side is already a perfect square trinomial. It can be written as:
(x - 3)^2 = 0
Taking the square root of both sides, we get:
x - 3 = 0
Adding 3 to both sides, we arrive at:
x = 3
Again, completing the square confirms our solution, x = 3.
Conclusion
In this exploration, we successfully navigated the quadratic equation x^2 = 6x - 9, employing factoring as our primary method to unveil the solution x = 3. We further validated our answer by demonstrating how the quadratic formula and completing the square would lead to the same result. This journey highlights the power and versatility of different techniques in solving quadratic equations, empowering us to tackle a wide range of mathematical challenges.
Find the value(s) of x in the quadratic equation x^2 = 6x - 9. Options: A) x=-3, x=2 B) x=0, x=1 C) x=3 D) x=9
Solving Quadratic Equations Find the Value of x in x^2 = 6x - 9
