Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(-3=4 x^2+9 x\)

Short answer

By using the quadratic formula, the solutions are \(x = \frac{-9 + \sqrt{33}}{8}\) and \(x = \frac{-9 - \sqrt{33}}{8}\). The Quadratic Formula method is preferred due to its efficiency and reliability.

Step-by-step solution

Rearrange Equation

First, rearrange the given quadratic equation to be equal to zero. Therefore, the equation becomes \(4x^2 + 9x + 3 = 0\)

Using Quadratic Formula

Next step, solve the rearranged equation with the Quadratic Formula, which is \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Here, a = 4, b = 9, and c = 3. By substituting the values of a, b, and c into the Quadratic Formula, the solution for the equation can be derived as \(x = \frac{-9 \pm \sqrt{81-48}}{8}\). Simplifying gives \(x = \frac{-9 \pm \sqrt{33}}{8}\). Thus, the roots of the equation are \(x = \frac{-9 + \sqrt{33}}{8}\) and \(x = \frac{-9 - \sqrt{33}}{8}\).

Using an Alternative Method (Factoring)

A preferable alternate method is factoring, but the equation \(4x^2 + 9x + 3 = 0\) is not simply factorable. Hence another method such as completing the square must be used. However, this can become a bit complicated. Looking at the complexity and time requirement of the second method, it can be concluded that using the Quadratic Formula is more efficient and reliable.

Key concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations of the format \(ax^2 + bx + c = 0\). It simplifies the process by yielding a formula that can be used directly:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The symbols \(a\), \(b\), and \(c\) represent the coefficients of the quadratic equation, extracted from \(ax^2 + bx + c = 0\). Here’s how to use it:

• Identify the coefficients \(a\), \(b\), and \(c\) from your equation.
• Plug these values into the Quadratic Formula.
• Calculate the discriminant, \(b^2 - 4ac\), to find out the nature of the roots.
• Simplify further to determine the values of \(x\).

The beauty of this method is its reliability; it works even when factoring is complicated or impossible.However, remember that if the discriminant (\(b^2 - 4ac\)) is negative, the equation has no real roots,but two complex roots.

Factoring Method

The Factoring method involves expressing the quadratic equation in a product of two binomials,generally aimed at finding two numbers that add up to \(b\) and multiply to \(ac\) if possible.Consider the quadratic \(ax^2 + bx + c = 0\), the goal is to identify specific factors that help in expressing it as \((px + q)(rx + s) = 0\).If a quadratic can be easily factored:

• Both roots can be directly obtained from the binomials.
• The sum of these factors needs to match \(b\), and their product should be \(ac\).

Although factoring can be a quicker solution for some equations, it isn't always feasible.Some quadratics, like in the given example, require more advanced methods because intuitive factoring isn’t possible when dealing with certain coefficients or when the quadratic cannot decompose nicely into binomials.

Completing the Square

Completing the square is a systematic method for solving quadratic equations.It involves rearranging the equation so that one side is a perfect square trinomial.The method provides a versatile approach for solving, even if factoring is not possible.Here's a simple breakdown:Start with the general quadratic formula \(ax^2 + bx + c = 0\):

• Move constant \(c\) to the other side of the equation.
• If \(a\) is not 1, divide the entire equation by \(a\).
• Add \((\frac{b}{2})^2\) to both sides to form a perfect square trinomial on the left.
• Express this trinomial as \((x + \frac{b}{2a})^2\).
• Solve for \(x\) by taking the square root of both sides.

Completing the square might require more steps than using the Quadratic Formula,but it’s a useful method for deriving the quadratics’ vertex form and has applications beyond just solving equations,into graphing and calculus scenarios.