Problem 66 - Algebra and Trigonometry

Question

Solve equation using the quadratic formula. $$ x^{2}+8 x+12=0 $$

Step-by-Step Solution

Answer

The solutions for the given quadratic equation are \(x_1 = -2\) and \(x_2 = -6\).

Step 1: Identify the coefficients

First, identify the coefficients as given in the equation, namely \(a=1\), \(b=8\), and \(c=12\).

Step 2: Calculate the discriminant

Next, calculate the discriminant (\(b^{2}-4ac\)). Substitute the coefficients into the formula: \(8^{2} - 4*1*12 = 64 - 48 = 16\).

Step 3: Applying the Quadratic Formula

Now we substitute the coefficients of \(a\), \(b\), and the discriminant into the quadratic formula to calculate \(x\): \[x=\frac{-b \pm \sqrt{b^{2}-4ac}} {2a} = \frac{-8 \pm \sqrt{16}}{2} \] This step simplifies to \(x_1 = -2\) (for the positive root of the discriminant) and \(x_2 = -6\) (for the negative root of the discriminant).

Key Concepts

Discriminant

The discriminant is a key element when solving a quadratic equation using the quadratic formula. It is given by the expression \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation of the form \( ax^2 + bx + c = 0 \).
The value of the discriminant tells us about the nature of the roots of the quadratic equation.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation has no real roots, but two complex roots.

In our exercise, the discriminant was calculated as 16, which is a positive number. This confirms that the quadratic equation has two distinct real roots. Understanding the outcome of the discriminant can help you predict the roots even before calculating them.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. Its general form is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
The solutions to this equation, known as the roots, can be found using various methods such as factoring, completing the square, or using the quadratic formula.
In the given exercise, the quadratic equation is \( x^2 + 8x + 12 = 0 \), where the coefficients are identified as \( a = 1 \), \( b = 8 \), and \( c = 12 \). These coefficients are crucial for calculating the discriminant and applying the quadratic formula to find the roots.
The quadratic equation often represents real-world situations such as projectile motion or optimizing areas and is a fundamental concept in algebra. By mastering the quadratic equation, you equip yourself with a versatile tool for solving a range of mathematical problems.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions for \( x \) when the equation is set to zero. There are several methods for finding these roots, but the quadratic formula is one of the most straightforward and universally applicable methods.
The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula uses the coefficients \( a \), \( b \), and \( c \) from the quadratic equation, as well as the discriminant \( b^2 - 4ac \). By substituting these into the formula, you can find the roots.

• The "\( \pm \)" symbol means there are generally two solutions or roots.
• The roots are often referred to as \( x_1 \) and \( x_2 \).

In our solution, the roots calculated using the quadratic formula are \( x_1 = -2 \) and \( x_2 = -6 \). These solutions align with a positive discriminant, indicating two distinct real roots. Knowing how to find these roots is crucial for solving quadratic equations and for applications in various fields such as physics and engineering.