Problem 21 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$4 x^{2}=2 x+7$$

Step-by-Step Solution

Answer

The solutions of the equation are \(x_{1} = \frac{1 - \sqrt{29}}{2}\) and \(x_{2} = \frac{1 + \sqrt{29}}{2}\).

Step 1: Arrange the Equation in Standard Quadratic Form

Rearrange the equation \(4x^{2} – 2x – 7 = 0\), where \(a = 4, b = -2\) and \(c = -7\).

Step 2: Calculate the Discriminant

Calculate the discriminant using the formula \(D = b^{2} – 4ac\). So, \(D = (-2)^{2} - 4*4*(-7) = 4 + 112 = 116\). The discriminant is positive, thus the equation has two real and distinct solutions.

Step 3: Find the Roots using Quadratic Formula

We substitute a, b, c and D into the quadratic formula. The solutions are given by \[x_{1} = \frac{{-(-2) - \sqrt{116}}}{2*4} = \frac{1 - \sqrt{29}}{2}\] and \[x_{2} = \frac{{-(-2) + \sqrt{116}}}{2*4} = \frac{1 + \sqrt{29}}{2}\].

Key Concepts

Solving Quadratic Equations

Quadratic equations are an essential part of algebra, featuring a specific form represented as \( ax^2 + bx + c = 0 \). Here, the coefficients are denoted as \( a \), \( b \), and \( c \), where \( a \) is not equal to zero. To solve such equations, the quadratic formula offers a robust method. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Applying this formula involves several steps:

• Ensure the equation is arranged in the standard quadratic form.
• Identify the values of \( a \), \( b \), and \( c \).
• Calculate the discriminant to find out the nature of the roots.
• Substitute the values into the quadratic formula to compute the solutions.

Breaking the problem into these steps can significantly aid in understanding and solving the equation effectively.

Discriminant Calculation

Calculating the discriminant is a crucial step in solving quadratic equations. The discriminant is derived from the formula \( b^2 - 4ac \), providing insight into the nature of the equation's solutions. Here is what the discriminant reveals:

• If the discriminant is greater than zero, the quadratic equation yields two distinct real roots. This implies the parabola intersects the x-axis at two points.
• If the discriminant is exactly zero, the equation has exactly one real root, meaning the parabola touches the x-axis at one tangent point.
• If the discriminant is less than zero, no real roots exist, indicating the parabola does not intersect the x-axis.

In the example provided, the discriminant calculation \( (-2)^2 - 4 \cdot 4 \cdot (-7) = 116 \) is positive, signifying two real and distinct solutions. This positive result is crucial for further solving the quadratic equation using the quadratic formula.

Real and Distinct Solutions

Solutions to a quadratic equation can be categorized based on the value of the discriminant. When the discriminant is positive, it guarantees two real and distinct solutions, as visibly seen in our quadratic formula.Upon calculating the discriminant and confirming its positive value, the next step involves substituting back into the quadratic formula. Given the formula:\[x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The positive and negative square roots lead to two solutions. In the given exercise:

• The first solution involves subtracting the square root: \( x_1 = \frac{1 - \sqrt{29}}{2} \)
• The second solution involves adding the square root: \( x_2 = \frac{1 + \sqrt{29}}{2} \)

These two distinct real numbers demonstrate how a positive discriminant indicates separate intersection points on the x-axis for the quadratic equation. Understanding these solutions allows us to interpret the graphical representation of the equation as well.