Problem 82 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 4 y^{2}-11 y-3=0 $$

Step-by-Step Solution

Answer

The solutions are \(y = 3\) and \(y = -\frac{1}{4}\).

Step 1: Identify the quadratic equation

The equation given is a quadratic equation in the form of \(ax^2 + bx + c = 0\). Here, \(a = 4\), \(b = -11\), and \(c = -3\). We're looking to solve for \(y\).

Step 2: Use the quadratic formula

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

Step 3: Calculate the discriminant

The discriminant of a quadratic equation \(b^2 - 4ac\) helps determine the nature of the roots. Calculate it as follows: \((-11)^2 - 4 \times 4 \times (-3) = 121 + 48 = 169 \). This means the discriminant is 169, which is a perfect square.

Step 4: Find the square root of the discriminant

Calculate \(\sqrt{169}\) to continue solving the equation. \(\sqrt{169} = 13 \).

Step 5: Apply the quadratic formula

Substitute the values into the quadratic formula: \( y = \frac{-(-11) \pm 13}{2 \times 4} = \frac{11 \pm 13}{8} \).

Step 6: Calculate the two potential solutions

Calculate each potential solution with \(+\) and \(-\):1. When using \(+\):\( y = \frac{11 + 13}{8} = \frac{24}{8} = 3 \).2. When using \(-\):\( y = \frac{11 - 13}{8} = \frac{-2}{8} = -\frac{1}{4} \).

Step 7: Present the solutions

The solutions to the quadratic equation \(4y^{2} - 11y - 3 = 0\) are \(y = 3\) and \(y = -\frac{1}{4}\).

Key Concepts

Discriminant

The discriminant is a key part of solving quadratic equations using the quadratic formula. It helps determine the nature and number of the roots of the equation. The discriminant is denoted by the expression \(b^2 - 4ac\), which comes from the quadratic formula.Here's why it's important:

• It indicates whether the roots are real or complex.
• It shows how many real roots there are.

In our example, the discriminant is 169, which is a perfect square. This perfect square suggests something interesting about the roots: they will both be real and rational. A perfect square discriminant implies that when you take the square root of it, you'll get a whole number, leading to simpler calculations. This helps ease the process of solving the quadratic equation and understanding the nature of its solutions.

Quadratic Formula

The quadratic formula is like a magic key for solving quadratic equations. It works for any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula itself is given by:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Let's break down how it works:

• The \(-b\) represents the opposite of the coefficient \(b\), flipping the sign.
• \(\sqrt{b^2 - 4ac}\) highlights the discriminant's role, as it tells us the nature of roots.
• The \(\pm\) sign shows that there are typically two solutions to consider — one with addition and one with subtraction.
• The denominator \(2a\) accounts for the stretch or shrink of the graph of the quadratic function, determined by \(a\).

So, for the equation \(4y^2 - 11y - 3 = 0\), plugging in values gives us our two potential solutions. This formula is versatile and powerful because it helps find the roots for any standard quadratic equation, no matter how complex it may seem.

Roots of the Equation

The roots of a quadratic equation are the solutions where the equation equals zero. Finding the roots is equivalent to finding where the graph of a quadratic equation crosses the x-axis.In our exercise, we solved \(4y^2 - 11y - 3 = 0\) and found two roots: \(y = 3\) and \(y = -\frac{1}{4}\). Here's what each root means:

• \(y = 3\): This is a point where the parabola intersects the y-axis and the quadratic expression equals zero. This illustrates that you have a zero or root at \(y = 3\).
• \(y = -\frac{1}{4}\): Similarly, this is another intersection point where the equation equals zero.

These roots tell us where the quadratic equation "cuts" the x-axis on its graph. Knowing these roots gives you a lot of information about the graph itself, such as symmetry and vertex position. Understanding roots is fundamental in algebra, as it connects equations and their geometric interpretations.