Problem 14 - 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. $$ 2 x^{2}=x+4 $$

Step-by-Step Solution

Answer

The roots are \(x = \frac{1 + \sqrt{33}}{4}\) and \(x = \frac{1 - \sqrt{33}}{4}\).

Step 1: Identify the standard form

The first step is to write the equation in the standard form of a quadratic equation, which is: \(ax^2 + bx + c = 0\). Our equation is \(2x^2 = x + 4\). To make it standard, we move all terms to one side: \(2x^2 - x - 4 = 0\).

Step 2: Identify coefficients

Next, we identify the values of \(a\), \(b\), and \(c\) from the standard form. In our equation \(2x^2 - x - 4 = 0\), we have \(a = 2\), \(b = -1\), and \(c = -4\).

Step 3: Use the quadratic formula

The quadratic formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We will use this formula to find the roots of the equation.

Step 4: Calculate the discriminant

The discriminant of a quadratic equation is \(b^2 - 4ac\). Substitute \(a = 2\), \(b = -1\), and \(c = -4\) into the formula: \((-1)^2 - 4 \cdot 2 \cdot (-4) = 1 + 32 = 33\). The discriminant is \(33\).

Step 5: Calculate the roots

Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{33}}{2 \cdot 2} = \frac{1 \pm \sqrt{33}}{4} \].

Step 6: Write the roots in simplest form

The roots of the equation, in simplest radical form, are \(x = \frac{1 + \sqrt{33}}{4}\) and \(x = \frac{1 - \sqrt{33}}{4}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in mathematics used to find the solutions, or "roots," of a quadratic equation. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \). Here, the coefficients \( a \), \( b \), and \( c \) are constants, with \( a \) not being zero. The quadratic formula allows us to determine the values of \( x \) by using the following formula:

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

This formula provides two solutions due to the "plus or minus" symbol (\( \pm \)). This means that we will perform two separate calculations to find the complete set of roots. Whether or not these roots are real depends on the value of the discriminant, which is the expression under the square root symbol.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is essential for applying the quadratic formula effectively. To express an equation in standard form, we rearrange all terms to one side of the equation to look like \( ax^2 + bx + c = 0 \). For instance, if you start with an equation such as \( 2x^2 = x + 4 \), you must rearrange it to align with the standard form by moving all terms to one side:

• Subtract \( x \) and \( 4 \) from both sides to yield \( 2x^2 - x - 4 = 0 \).

Once in standard form, it becomes straightforward to identify the coefficients \( a \), \( b \), and \( c \), which are crucial for using the quadratic formula. This organization is vital because it allows for uniform application of the quadratic formula to find the roots.

Discriminant

The discriminant is a component of the quadratic formula that provides vital information about the nature of the roots of the quadratic equation. It is given by the expression \( b^2 - 4ac \). Depending on its value, the discriminant can tell us the following about the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a "repeated" root.
• If the discriminant is negative, the roots are complex or imaginary, meaning no real number solutions exist.

In the example \( 2x^2 - x - 4 = 0 \), the discriminant computes to \( 33 \), which is positive. Therefore, this equation has two distinct real roots.

Simplest Radical Form

When solving quadratic equations, especially those with irrational roots, expressing your answers in the simplest radical form can provide clearer insight into their true nature. Simplest radical form means simplifying the square root if possible, such that there are no square roots left in the denominator, and all radicands are in their simplest form. Here’s how that works:For the equation in the example, the solution using the quadratic formula led us to roots \( \frac{1 + \sqrt{33}}{4} \) and \( \frac{1 - \sqrt{33}}{4} \). These roots are already in their simplest radical form because:

• The square root term \( \sqrt{33} \) cannot be further simplified as 33 is not a perfect square and has no smaller square factor.

Thus, presenting the roots in this form avoids decimals and keeps them in an accurate, precise manner that reflects the exact solutions.