Problem 27 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(x^2-10 x-4=0\)

Step-by-Step Solution

Answer

The solutions to the given equation \(x^2-10x-4=0\) are \(x = 5 + \sqrt{29}\) and \(x = 5 - \sqrt{29}\).

Step 1: Identify a, b, and c

In the given equation, \(x^2-10x-4=0\), the coefficients are as follows: \(a = 1, b = -10\), and \(c = -4\).

Step 2: Plug a, b and c into the quadratic formula

Substitute \(a, b\), and \(c\) into the quadratic formula, which gives: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), so, \(x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 * 1 * -4}}{2 * 1}\).

Step 3: Simplify the expression

Simplify the expression, which results in: \(x = \frac{10 \pm \sqrt{100+16}}{2}\), or \(x = \frac{10 \pm \sqrt{116}}{2}\), or \(x = \frac{10 \pm 2\sqrt{29}}{2}\). This simplifies further to \(x = 5 \pm \sqrt{29}\).

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is foundational in algebra. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\). The solution to these equations can be found using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). This formula provides the values of \(x\) that satisfy the equation, which are called 'roots'.

To apply the quadratic formula, you'll first identify the coefficients (\(a\), \(b\), and \(c\)) from your quadratic equation. Once these are known, you simply plug them into the formula and simplify. In some cases, you might end up with a positive and a negative root, known as 'real and distinct' solutions; one real solution, known as a 'real and repeated' solution; or complex solutions that include imaginary numbers if the discriminant (\(b^2-4ac\)) is negative. When the discriminant is zero, it signals that both roots are equal.

Coefficients in Quadratic Equations

The coefficients in a quadratic equation are the numerical parts that multiply the variable \(x\). These coefficients are crucial as they determine the shape and position of the parabola when the quadratic equation is graphed. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, \(\textbf{a}\) is the coefficient of \(x^2\) and affects the width and direction of the parabola. Coefficient \(\textbf{b}\) is the linear coefficient and affects the symmetry and where the parabola crosses the \(y\)-axis, while \(\textbf{c}\) is the constant term which tells us the parabola's intersection with the \(y\)-axis.

Understanding the roles of these coefficients can greatly aid in graphing the function and predicting the nature of its roots. For instance, if \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, the parabola opens downwards. Adjusting these values can shift the graph left, right, up, or down.

Simplifying Radical Expressions

When solving quadratic equations using the quadratic formula, you may encounter radical expressions, which are expressions that contain a square root, cube root, or higher. Simplifying radical expressions is an important part of arriving at the most simplified solution for a quadratic equation. To simplify a radical expression, look for factors of the number under the radical that are perfect squares.

For example, let's consider the expression \(\sqrt{116}\). You can factor 116 into \(4 \times 29\), and realize that 4 is a perfect square. The radical expression simplifies to \(2\sqrt{29}\), since \(\sqrt{4} = 2\). Continue to look for such opportunities to simplify radicals to their smallest possible positive factor. Remember, this part of the process can dramatically change the appearance, but not the value, of the final expression. Simplifying radical expressions makes them easier to understand and work with, especially when you're performing further algebraic operations.