Problem 8 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ a^{2}+7 a+12=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(a^2 + 7a + 12 = 0\) by completing the square. Answer: The solutions for the given quadratic equation are \(a_1 = -3\) and \(a_2 = -4\).

Step 1: Divide the equation by the leading coefficient

Since the coefficient of the \(a^2\) term is already 1, we don't need to do anything for this step. We move on to the next step.

Step 2: Rewrite the quadratic equation by adding and subtracting the square of half of the linear coefficient

We have \(a^2 + 7a + 12 = 0\). Take half of the linear coefficient (7) which is \(\frac{7}{2}\). The square of \(\frac{7}{2}\) is \(\left(\frac{7}{2}\right)^2 = \frac{49}{4}\), so add and subtract this value from our equation to maintain equality: $$ a^2 + 7a + \frac{49}{4} - \frac{49}{4} + 12 = 0 $$

Step 3: Rewrite the expression as a square

The left side of the equation is now a perfect square trinomial and can be rewritten as the square of a binomial: $$ \left(a + \frac{7}{2}\right)^2 - \frac{49}{4} + 12 = 0 $$

Step 4: Simplify the expression and isolate the square

Simplify by combining constants on the right side: $$ \left(a + \frac{7}{2}\right)^2 = \frac{49}{4} - 12 $$ Since \(12 = \frac{48}{4}\), we can rewrite it as: $$ \left(a + \frac{7}{2}\right)^2 = \frac{49 - 48}{4} = \frac{1}{4} $$

Step 5: Take the square root of both sides

Now, we can take the square root of both sides to solve for a: $$ a + \frac{7}{2} = \pm\sqrt{\frac{1}{4}} $$

Step 6: Solve for the variable

To get the values of a, subtract \(\frac{7}{2}\) from both sides: $$ a = - \frac{7}{2} \pm \sqrt{\frac{1}{4}} $$ Since the square root of \(\frac{1}{4}\) is \(\frac{1}{2}\), we can rewrite the equation as: $$ a = - \frac{7}{2} \pm \frac{1}{2} $$ Now we solve for a by adding and subtracting \(\frac{1}{2}\): $$ a_1 = - \frac{7}{2} + \frac{1}{2} = - \frac{6}{2} = -3 \\ a_2 = - \frac{7}{2} - \frac{1}{2} = - \frac{8}{2} = -4 $$ So the solutions for the given quadratic equation are \(a_1 = -3\) and \(a_2 = -4\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where the variable \(x\) is raised to the second power or squared, and \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The graph of a quadratic equation forms a parabola, which can open upwards or downwards depending on the sign of \(a\).

These equations have various methods of solutions, including factoring, using the quadratic formula, and completing the square. Solving them involves finding the values of \(x\) that make the equation true. For the exercise at hand, we are focusing on completing the square, a method that transforms the equation into a perfect square trinomial, making it easier to solve.

Understanding quadratic equations is essential because they appear frequently in mathematical problems and applications in various fields such as physics and engineering. Recognizing the standard form and knowing different solution strategies can simplify working with them.

Solving Equations

Solving equations usually involves finding the value of the variable that satisfies the equation. In the context of quadratic equations, solving means determining the value of the unknown that will satisfy the equation \(ax^2 + bx + c = 0\).

The method of completing the square involves transforming the quadratic equation into the form \((x - p)^2 = q\), which can then be easily solved by taking the square root of both sides. This method is especially useful when the equation does not factor easily or when using other methods like the quadratic formula might be cumbersome for mental calculations or exams.

The steps include:

• Ensuring the coefficient \(a\) of the \(x^2\) term is 1. If it's not, divide the entire equation by \(a\).
• Adding and subtracting the square of half the linear coefficient \(b\) to and from the equation.
• Rewriting the quadratic term and the newly added value as a perfect square.
• Solving the resulting simpler equation for its roots by isolating the variable and applying the square root.

Practicing these steps helps build proficiency and flexibility in solving quadratic equations using various methods.

Algebraic Techniques

Algebraic techniques are various methods used to manipulate and solve mathematical expressions and equations. In the realm of quadratic equations, they offer a systematic pathway to finding solutions. Completing the square is one of these algebraic techniques, allowing for the transformation of incomplete trinomials into perfect squares.

Why use completing the square? This approach is particularly effective because:

• It can always solve any quadratic equation, regardless of whether it can be factored easily or not.
• It offers a deeper understanding of the properties of parabolic equations and their vertex form.
• It brings into perspective the symmetry of a parabola by identifying its vertex.

Steps in the technique are systematic, focusing on creating a perfect square trinomial that matches the form \((x + d)^2\), where \(d\) is derived from half the linear coefficient.

The consistent practice of algebraic techniques ensures that equations are approached methodically and with confidence. These skills are invaluable, underpinning more advanced mathematics and careers in STEM fields.