Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(3 x^2-21=3\)

Short answer

The solutions using the quadratic formula are \(x = -2, 2\), and the solutions using the completing the square method are \(x = -2sqrt{2}, 2sqrt{2}\). The Quadratic Formula Method seems simpler for this specific problem because it provided the answer more straightforwardly without the need for additional steps.

Step-by-step solution

Arrange the Equation

First, arrange the equation into standard quadratic form \(ax^2 + bx + c = 0\) by subtracting 3 from both sides. The resulting equation is \(3x^2 - 24 = 0\).

Solving through Quadratic Formula

The quadratic formula is \(x = [ -b ± sqrt(b^2 - 4ac) ] /2a\). With \(a = 3, b = 0 \) and \( c = -24\), the solutions are \(x = [ -0 ± sqrt((0)^2 - 4*3*(-24)) ] /2*3\), which simplifies to \(x = ±2\).

Solving through Completing the Square

The equation \(3x^2 - 24 = 0\) can be solved by completing the square. First, divide the equation by 3 to get \(x^2 - 8 = 0\). Add 8 to both sides to isolate the squares: \(x^2 = 8\). Then, take the square root of both sides: \(x = ± sqrt{8}\), which simplifies to \(x = ±2sqrt{2}\).

Determining Preferred Method

The quadratic formula gave the result more straightforwardly in this case because there was no \(b\) term. Completing the square required the additional step of square rooting. Thus, the first method was simpler for this specific equation.

Key concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations, which are equations in the form of (ax^2 + bx + c = 0). It is particularly useful because it provides a direct solution for the values of x, no matter the coefficients involved.

Using the formula (x = [ -b ± sqrt(b^2 - 4ac) ] / (2a)), you're able to find the roots of any quadratic equation by substituting the coefficients a, b, and c into the formula. To illustrate, using the example provided: (3x^2 - 21 = 3), you would first rearrange the equation to (3x^2 - 24 = 0). Here, a = 3, b = 0 (since the bx term is missing), and c = -24.

When these values are plugged into the Quadratic Formula, the solution simplifies to x = ±2, giving us two possible solutions. This method is widely appreciated for its clarity and straightforwardness, especially in scenarios where the b term equals zero, simplifying the calculation and making it the most efficient solution.

Completing the Square

The method of Completing the Square is another classical approach for solving quadratic equations. It involves converting a quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root of both sides.

Consider again the equation (3x^2 - 24 = 0). To use Completing the Square, start by ensuring the x^2 term has a coefficient of one. If not, you divide the entire equation by the coefficient of x^2, in this case, 3, resulting in (x^2 - 8 = 0).

Next, isolate the x^2 term by adding 8 to both sides, yielding (x^2 = 8). Taking the square root of both sides gives you x = ±sqrt(8), which further simplifies to x = ±2sqrt(2). While this method is elegant and works well in cases where the quadratic is not easily factorable, it can be more cumbersome than using the Quadratic Formula, especially if the equation does not readily transform into a perfect square.

Quadratic Equation Standard Form

The standard form of a quadratic equation is crucial for applying most solution techniques, including the Quadratic Formula and Completing the Square. It is expressed as (ax^2 + bx + c = 0), where a, b, and c are constants, and a eq 0, as an equation with a = 0 no longer remains quadratic.

In the exercise provided, the equation (3x^2 - 21 = 3) must first be converted into the standard form. This is achieved by subtracting 3 from both sides to yield (3x^2 - 24 = 0). With the equation in standard form, it becomes clear that the a coefficient is 3, there's no b term, and c is -24, which allows for the application of various solving techniques.

Understanding standard form is essential for identifying the coefficients and effectively analyzing the nature of the roots. For instance, the presence or absence of the b term can significantly influence the method choice for solving the equation. It's important to grasp this structure to successfully tackle quadratic equations using the most appropriate and efficient methods.