Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ 5 x+1=2 x^{2} $$

Step-by-Step Solution

Answer

The roots are \( x = \frac{5 + \sqrt{33}}{4} \) and \( x = \frac{5 - \sqrt{33}}{4} \).

Step 1: Write the equation in standard form

First, move everything to one side of the equation to form a standard quadratic equation. Start with the given equation: \( 5x + 1 = 2x^2 \). Subtract \( 5x \) and \( 1 \) from both sides to get: \( 2x^2 - 5x - 1 = 0 \).

Step 2: Identify the coefficients

In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 2 \), \( b = -5 \), and \( c = -1 \).

Step 3: Write the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use these coefficients: \( a = 2 \), \( b = -5 \), and \( c = -1 \).

Step 4: Calculate the discriminant

The discriminant \( b^2 - 4ac \) determines the nature of the roots. Calculate it: \( (-5)^2 - 4(2)(-1) = 25 + 8 = 33 \).

Step 5: Apply the quadratic formula

Substitute the values into the quadratic formula: \( x = \frac{-(-5) \pm \sqrt{33}}{2(2)} \). Simplify: \( x = \frac{5 \pm \sqrt{33}}{4} \).

Step 6: Interpret the result

The solutions to the quadratic equation \( 2x^2 - 5x - 1 = 0 \) are \( x = \frac{5 + \sqrt{33}}{4} \) and \( x = \frac{5 - \sqrt{33}}{4} \). These are the roots in simplest radical form.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation. It's especially helpful when factoring isn't simple. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are the coefficients of the terms in the quadratic equation \(ax^2 + bx + c = 0\).

• \(x\) represents the solution(s) or roots of the equation.
• "\(\pm\)" means "plus or minus", indicating that typically two solutions may be possible.

So, by plugging in these values into the formula, you can solve for \(x\), which reveals the roots of the quadratic equation. Remember, the formula works for all quadratic equations, whether they can be factored or not.

Standard Form

Before you use the quadratic formula, you must ensure that your equation is in the standard form of a quadratic equation: \(ax^2 + bx + c = 0\). This makes it easier to identify the coefficients \(a\), \(b\), and \(c\) which are crucial for using the quadratic formula.

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

In the example \(5x + 1 = 2x^2\), we rearrange it to \(2x^2 - 5x - 1 = 0\). This step is crucial because applying the formula requires these coefficients to be clear and in this specific order to accurately use them in the quadratic formula.

Discriminant

The discriminant is an essential part of the quadratic formula. It is within the square root part of the formula: \(\sqrt{b^2 - 4ac}\) and is denoted by \(D = b^2 - 4ac\). The discriminant provides valuable information about the nature of the roots of the quadratic equation.

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, also known as a repeated or double root.
• If \(D < 0\), the roots are not real numbers; they are complex or imaginary roots.

In our example, \(b^2 - 4ac = 33\), which is greater than zero, indicating two distinct real roots. Therefore, there are real solutions for \(x\) in this equation.

Roots

Roots, also called solutions or zeroes, are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). In quadratic equations, roots can be found using various methods, such as factoring, completing the square, or using the quadratic formula.
In the context of our example, the roots were found using the quadratic formula, leading to:

• \(x = \frac{5 + \sqrt{33}}{4}\)
• \(x = \frac{5 - \sqrt{33}}{4}\)

These represent the values of \(x\) where the original equation equals zero. Remember, roots are the x-values that tell us where a quadratic equation touches or crosses the x-axis in a graph. In this case, because the discriminant is positive, both roots are real and distinct.