Problem 20 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$4 x^{2}+12 x+9=0$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = -1.5\).

Step 1: Identify Parameters a, b, c

Given equation is \(4x^{2} + 12x + 9 = 0\). Compare this with \(ax^{2} + bx + c = 0\) to identify the parameters as a=4, b=12, and c=9.

Step 2: Apply the Quadratic Formula

Plugging these values into the quadratic formula gives us: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} = \frac{{-12 \pm \sqrt{{(12)^2 - 4*4*9}}}}{2*4}\].

Step 3: Simplify the expression

The radical in the quadratic formula simplifies as \[\sqrt{{(12)^2 - 4*4*9}} = \sqrt{0}\]. Substituting this back into the formula gives us \[x = \frac{{-12 \pm 0}}{2*4}\]

Step 4: Compute the solution

Finally, simplifying the above gives \(x = -1.5\).

Key Concepts

Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula provides the roots of the equation, which are the values of \(x\) that satisfy the equation. The quadratic formula is given by:

\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]

Here's how it works:

• Identify constants \(a\), \(b\), and \(c\) from the equation.
• Use these constants in the formula to find the roots.
• The symbol \(\pm\) indicates that there may be two solutions: one with addition and one with subtraction.

The portion under the square root, \(b^2 - 4ac\), is called the discriminant, which determines the number and type of solutions.

If it is positive, there are two distinct real roots. If it's zero, there is one real root. For a negative discriminant, the equation has no real solutions, indicating complex roots.

Polynomial Equations

Polynomial equations consist of variables and coefficients arranged in terms of powers. These equations can involve one or more terms or powers. Quadratic equations are a specific type of polynomial equations where the highest power of the variable is 2. Here is how we identify and work with them:

• Typically, polynomial equations are in the form \(a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 = 0\).
• The highest exponent, like \(2\) in a quadratic, determines the equation type—linear (1), quadratic (2), cubic (3), etc.

Quadratic equations, like the one given in this problem \(4x^2 + 12x + 9 = 0\), have a standard form \(ax^2 + bx + c = 0\). They are solved by factoring, completing the square, or applying the quadratic formula as shown.

Solving polynomial equations helps in finding real-valued solutions and understanding relationships between variables, crucial in many fields such as physics, engineering, economics, and beyond.

Irrational Solutions

Irrational solutions are the roots of an equation that cannot be expressed as a simple fraction. These solutions often involve square roots of numbers that are not perfect squares. In many cases, you'll encounter irrational numbers when solving quadratic equations using the quadratic formula.

For instance, given \(b^2 - 4ac\) results in a number that is not a perfect square, the solution will include a square root that cannot be simplified to an integer or a fraction.

However, some irrational numbers can sometimes be expressed in a simplified radical form. In our exercise, the discriminant is zero, which simplifies to \(\sqrt{0}\), leading to a rational solution, \(x = -1.5\). But in cases with non-zero discriminants resulting in irrational numbers, you can use methods to simplify such radicals if the number inside the square root can be broken down into a product involving a perfect square.