Problem 90 - Algebra 2

Question

Solve using the Quadratic Formula. \(x^{2}+10 x+11=0\)

Step-by-Step Solution

Answer

The solution to the quadratic equation \(x^2 + 10x + 11 = 0\) is \(x = -5 ± \sqrt{14}\)

Step 1: Identifying Coefficients

Identify coefficients from the given quadratic equation \(x^2 + 10x + 11 = 0\), where \(a = 1\), \(b = 10\), and \(c = 11\)

Step 2: Apply the Quadratic Formula

Substitute the values of \(a\), \(b\), and \(c\) into the Quadratic Formula (1) to get \(-10 ± \sqrt{{10}^2 - 4 * 1 * 11}\) / \(2 * 1\) which simplifies to \(-10 ± \sqrt{100 - 44}\) / \(2\)

Step 3: Calculate and Simplify

Calculate under the square root and simplify further to get \(-10 ± \sqrt{56}\) / \(2\), which becomes \(-10 ± 2 \sqrt{14}\) / \(2\)

Step 4: Final Answer

Simplify to get the roots as \(-5 ± \sqrt{14}\)

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). These equations have a characteristic curve called a parabola. To solve quadratic equations, we aim to find the values of \(x\) that satisfy this equation. One popular and powerful method to do this is through the Quadratic Formula.

The Quadratic Formula provides a direct solution to the quadratic equation and is especially useful when the equation cannot be easily factored. The formula is given by:

• \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)

Using this formula, you can find the roots (or solutions) of any quadratic equation by substituting the coefficient values from the equation. It offers a reliable means to solve for \(x\) even when other methods are cumbersome. The key is in substituting accurately and carefully calculating the resulting expression.

This method is effective for all quadratic equations as long as you are careful with your calculations, especially when dealing with the square root and fractions. When using this formula, remember the “±” symbol indicates that there will be two solutions, corresponding to the top and bottom branches of the parabola.

Quadratic Equation Coefficients

Coefficients play a crucial role in understanding and solving quadratic equations. In the standard form of a quadratic equation, \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are known as the coefficients.

Each coefficient has a specific job:

• \(a\) is the coefficient of the \(x^2\) term, known as the leading coefficient. It determines the width and direction of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \(b\) is the coefficient of the \(x\) term. It affects the position of the vertex of the parabola along the x-axis.
• \(c\) is the constant term, which influences the location of the parabola along the y-axis.

Identifying these coefficients correctly is the first step in solving quadratic equations using the Quadratic Formula. In our example problem \(x^2 + 10x + 11 = 0\), the coefficients were identified as \(a = 1\), \(b = 10\), and \(c = 11\). Correctly recognizing these values ensures accurate substitution into the formula, leading to the correct roots.

Quadratic Root Calculation

Once the coefficients \(a\), \(b\), and \(c\) are identified, the next step is the calculation of the roots using the Quadratic Formula. The process involves several careful calculations:

First, plug the coefficients into the formula:

• \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)

For our example, substituting \(a = 1\), \(b = 10\), and \(c = 11\), the expression becomes:

• \(x = \frac{ -10 \pm \sqrt{10^2 - 4 \times 1 \times 11} }{2 \times 1}\)
• Simplify under the square root: \(100 - 44 = 56\)
• So, \(x = \frac{ -10 \pm \sqrt{56} }{2}\)

Next, simplify the square root. If possible, express it in simplest radical form. Here, \(\sqrt{56} = 2\sqrt{14}\). Further simplifying gives:

• \(x = \frac{ -10 \pm 2\sqrt{14} }{2}\)

Finally, divide the terms by 2 to get the roots:

• \(x = -5 \pm \sqrt{14}\)

These roots represent the points where the parabola crosses the x-axis. Understanding how to perform each calculation step by step ensures that you arrive at accurate solutions when using the Quadratic Formula.