Problem 3 - Algebra 2

Question

Find the exact solutions by using the Quadratic Formula. \(4 x^{2}+4 x+1=0\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\).

Step 1: Identify Quadratic Equation Coefficients

The given quadratic equation is in the form of \(ax^2 + bx + c = 0\). Here, identify the coefficients: \(a = 4\), \(b = 4\), and \(c = 1\).

Step 2: Recall the Quadratic Formula

The quadratic formula for finding the roots of a quadratic equation \(ax^2 + bx + c = 0\) is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) using the identified coefficients: \(b^2 - 4ac = 4^2 - 4 \times 4 \times 1\). Perform calculations: \(16 - 16 = 0\).

Step 4: Apply the Quadratic Formula

Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula. Since the discriminant is 0, the ± term becomes irrelevant: \(x = \frac{-4 \pm \sqrt{0}}{8} = \frac{-4}{8}\).

Step 5: Simplify the Solution

Simplify the fraction \(\frac{-4}{8}\) to find the solution: \(x = -\frac{1}{2}\).

Key Concepts

Discriminant

The discriminant is a fundamental component when using the quadratic formula to solve quadratic equations. It is the part under the square root in the quadratic formula: \(b^2 - 4ac\). This value can tell us a lot about the nature of the roots of the quadratic equation. Let's dive into what these values mean:

• If the discriminant is positive, \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots. The square root of a positive number is a real number, allowing for two solutions when considering the plus-minus component of the quadratic formula.
• If the discriminant equals zero, \(b^2 - 4ac = 0\), there is exactly one real root, known as a repeated or double root. This occurs because the square root of zero is zero, resulting in the formula simplifying to just \(x = \frac{-b}{2a}\).
• If the discriminant is negative, \(b^2 - 4ac < 0\), the quadratic equation has no real roots but instead two complex roots. Complex numbers arise because the square root of a negative number is imaginary.

In our example \(4x^2 + 4x + 1 = 0\), the discriminant is calculated as \(16 - 16 = 0\), indicating there is one repeated real root.

Coefficients of a quadratic equation

Quadratic equations are polynomial equations of degree two, typically taking the form \(ax^2 + bx + c = 0\). Here, the letters \(a\), \(b\), and \(c\) represent the coefficients and constant of the equation:

• \(a\) is the coefficient of the quadratic term \(x^2\). It ensures the equation is quadratic. If \(a = 0\), the equation is no longer quadratic but linear.
• \(b\) is the coefficient of the linear term \(x\). It influences the slope and direction of the parabola's axis.
• \(c\) is the constant term. It affects the vertical shift of the parabola on a graph.

In our given problem, \(4x^2 + 4x + 1 = 0\), we identified the coefficients as \(a = 4\), \(b = 4\), and \(c = 1\). Each plays a crucial role in shaping the equation and determining the parabola's graph.

Roots of a quadratic equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be real or complex, depending on the discriminant. The primary method to find these roots is using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).In our example, after calculating the discriminant and determining it to be zero, we simplified the quadratic formula. Without the ± component (since the square root of zero is zero), it becomes:\[x = \frac{-4}{8} = -\frac{1}{2}\]Thus, the root is \(-\frac{1}{2}\). This type of solution, where both roots are the same, is also visually represented by the parabola touching the x-axis at this single point.