Problem 6 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$x^{2}+12 x+36=0$$

Step-by-Step Solution

Answer

The solution of the quadratic equation \(x^2 +12x + 36 = 0\) using the quadratic formula is \(x = -6\).

Step 1: Identify the coefficients

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). From the given equation \(x^2 +12x + 36 = 0\), we can identify the coefficients as follows: a = 1, b = 12, and c = 36.

Step 2: Insert the coefficients into the quadratic formula

We substitute a, b, c into the formula to find: \(x = \frac{-12 \pm \sqrt{12^2 - 4*1*36}}{2*1}\).

Step 3: Simplify under the square root

Before we can solve the equation, we need to simplify what's under the square root. The calculation becomes \(x= \frac{-12 \pm \sqrt{144 - 144}}{2}\).

Step 4: Solve the equation

We're left with \(x = \frac{-12 \pm \sqrt{0}}{2}\), which simplifies to \(x = \frac{-12 \pm 0}{2} = -6\). Therefore, the equation has one solution, \(x = -6\) .

Key Concepts

Solving Quadratic Equations

When you come across an equation like \(x^2 +12x + 36=0\), you're dealing with a quadratic equation, characterized by the highest exponent being 2. To solve such equations, you can't just isolate the variable as you would in a linear equation. Among several methods available, the quadratic formula provides a systematic approach, ideal for handling any form that can be rewritten as \(ax^2 + bx + c = 0\). This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a direct path to finding the roots of the equation, where roots are the values of \(x\) that satisfy the equation.

Let's briefly go over how the formula works. After identifying your coefficients, you substitute them into the formula, perform the operations under the square root (known as the discriminant), and finally, simplify the fraction to find the values of \(x\). In the exercise given, we applied these steps methodically to find the single solution to the quadratic equation.

Quadratic Equation Coefficients

Understanding the roles of coefficients in a quadratic equation is fundamental to using the quadratic formula effectively. In any quadratic equation of the form \(ax^2 + bx + c = 0\), 'a' represents the quadratic coefficient, 'b' is the linear coefficient, and 'c' is the constant term.

The values of these coefficients directly influence the nature and number of solutions to the equation. For example, if you change the value of 'a', the parabola's width and direction alter, affecting where it might intersect the x-axis. Coefficients 'b' and 'c' impact the location of the vertex and the parabola's height. In our exercise, the coefficients are 1, 12, and 36, which made it straightforward to insert into the quadratic formula after identification.

Simplifying Square Roots

A crucial step in the quadratic formula is simplifying the square root part, the discriminant \(\sqrt{b^2-4ac}\). Simplifying the square root helps us determine the number and type of solutions. If the discriminant is positive, we get two real solutions; if zero, one real solution; and if negative, two complex solutions.

In the example exercise, we find the discriminant to be \(\sqrt{144 - 144} = \sqrt{0}\), which simplifies to 0. This means we have only one real solution when substituting back into the quadratic formula. Simplification of the square root is a place where attention to detail is crucial. Mistakes here can easily lead to incorrect solutions, which is why it's beneficial for learners to master square root properties and simplification techniques.