Question

The class is asked to solve the equation \(4 x^2+14 x+11=0\). You decide to solve the equation by completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Explain your reasoning.

Short answer

Both completing the square and Quadritic Formula methods correctly solve the quadratic equation, obtaining solutions \(x=-1/2\) and \(x=-3\). However, the Quadratic Formula is generally more efficient as it is applicable universally with less rearranging and calculating required.

Step-by-step solution

Solve by Completing the Square

Write the equation in the form of \(ax^2+bx+c=0\). Identify a, b, and c from the original equation: \(a=4\), \(b=14\), and \(c=11\). Rearrange the equation by isolating the quadratic terms on one side of the equation, so it will become, \(4x^2+14x=-11\). Then, convert the left side of the equation to a perfect square. To achieve this, divide b by 2 and square the result and add it both sides of the equation. Hence, the equation becomes, \((2x+7)^2 = 36\). Finally, find the square root of both sides, and solve for x, which gives \(x= -1/2\) or \(x=-3\).

Solve by using the Quadratic Formula

For the quadratic equation in the form of \(ax^2 + bx + c = 0\), the solutions can be obtained using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here a is 4, b is 14 and c is 11. Substitute these values into the formula to find the solutions. \(x = \frac{-14 \pm \sqrt{14^2-4*4*11}}{2*4}\), which simplifies to \(x = -1/2\) or \(x=-3\).

Compare the Efficiency of the Two Methods

Both methods have given the same solutions which means both are correct. However, the complete the square method requires rearrangement of the equation, and finding a number to complete the square which can be complicated for some equations. On the other hand, the Quadratic Formula is straightforward and can be applied universally without having to adjust the equation first. Therefore, according to this comparison, the Quadratic Formula tends to be a more efficient method to solve quadratic equations.

Key concepts

Completing the Square

Completing the square is a traditional method used to solve quadratic equations. It involves rearranging the equation so that a perfect square trinomial is formed. Let's break down what that means:

• Start with a quadratic equation in the form: \(ax^2 + bx + c = 0\).
• Rearrange it so that only the terms involving \(x\) are on one side: \(ax^2 + bx = -c\).
• Divide all terms by \(a\) (if \(a eq 1\)) to make the leading coefficient 1.
• The key step is to take the coefficient \(b\), divide it by 2, and square the result. This turns the quadratic expression into a perfect square.
• Add this squared value to both sides to maintain equality.
• Express the quadratic expression as a squared binomial and solve for \(x\).

In our exercise, this method transformed the equation \(4x^2 + 14x + 11 = 0\) into \((2x + 7)^2 = 36\), solving gives \(x=-1/2\) or \(x=-3\). Although it can be a bit involved, completing the square helps in understanding the geometry of parabolas and is also pivotal when studying algebraic identities.

Quadratic Formula

The Quadratic Formula provides a straightforward method for finding roots of any quadratic equation. It is particularly useful because it does not require any modifications to the equation beyond identifying \(a\), \(b\), and \(c\) from \(ax^2 + bx + c = 0\). The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• This formula directly gives the solutions by substituting the values of \(a\), \(b\), and \(c\).

For our quadratic \(4x^2 + 14x + 11 = 0\), substituting gives \(x = -1/2\) or \(x = -3\). The advantage of the Quadratic Formula is its universality and speed; it quickly provides roots without additional steps. Moreover, the discriminant \(b^2 - 4ac\) reveals information about the nature of the roots: whether they are real or complex, and if they are distinct or repeated.

Efficiency of Methods

When solving quadratic equations, efficiency is often a matter of context and simplicity. Comparing methods in our exercise, both completing the square and using the quadratic formula give the same results, but the paths differ:

• Completing the Square involves analysis of coefficients, rearrangement, and sometimes complex arithmetic, which may be a multi-step process.
• Quadratic Formula is typically faster, requiring only substitution into the formula, making it less prone to error, especially with complex equations.

For most students, the Quadratic Formula is more efficient due to its straightforward nature and applicability to all quadratic equations without adjustments. However, mastering both methods is valuable, as completing the square provides deeper insights into quadratic expressions and their properties.