Problem 71 - Elementary Algebra

Question

Solve using the quadratic formula. $$ 3 y 2-2 y+4=0 $$

Step-by-Step Solution

Answer

The equation has two complex roots: \( y = \frac{1 + i \sqrt{11}}{3} \) and \( y = \frac{1 - i \sqrt{11}}{3} \).

Step 1: Identify the coefficients

The quadratic equation is given in the form \( ay^2 + by + c = 0 \). In this equation, \( a = 3 \), \( b = -2 \), and \( c = 4 \).

Step 2: Write down the quadratic formula

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

Step 3: Calculate the discriminant

Compute the discriminant, \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \): \((-2)^2 - 4(3)(4) = 4 - 48 = -44\).

Step 4: Determine the nature of the roots

Since the discriminant is negative (\(-44\)), it indicates that the quadratic equation has no real roots, only complex roots.

Step 5: Compute the complex roots

Substitute \( a = 3 \), \( b = -2 \), and the discriminant into the quadratic formula: \( y = \frac{-(-2) \pm \sqrt{-44}}{2(3)} = \frac{2 \pm \sqrt{-44}}{6} \). This simplifies to: \( y = \frac{2 \pm i\sqrt{44}}{6} \). Further simplify: \( y = \frac{2 \pm i 2\sqrt{11}}{6} \).Simplify again: \( y = \frac{1 \pm i \sqrt{11}}{3} \).

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula. It is found within the square root part of the formula: \( b^2 - 4ac \). The discriminant tells us about the nature of the roots of a quadratic equation:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there are exactly one real root (a repeated root).
• If the discriminant is negative, the quadratic equation has no real roots but two complex roots.

In our example, the discriminant was calculated as \(-44\). This negative result clearly indicates that the equation will not intersect the x-axis, resulting in complex roots.

Complex Roots

Complex roots occur when the discriminant of a quadratic equation is negative. In such cases, the roots are not real numbers and instead include imaginary numbers.This is seen when the square root of a negative number is involved, introducing the imaginary unit, \(i\), where \(i = \sqrt{-1}\). In the given exercise, after substituting the coefficients into the quadratic formula, we ended up with the roots: \(y = \frac{1 \pm i \sqrt{11}}{3}\). These roots are a combination of a real part \(\frac{1}{3}\) and an imaginary part \(\frac{i \sqrt{11}}{3}\).Understanding complex roots is essential, as they frequently appear in both mathematics and engineering disciplines.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that can be expressed in the form \(ay^2 + by + c = 0\). It includes:

• Quadratic term: the \(ay^2\) part.
• Linear term: the \(by\) part.
• Constant term: the \(c\) part.

These equations are notable for their parabolic graphs and are a foundational component of algebra. To solve a quadratic equation, one may employ various methods such as factoring, completing the square, or using the quadratic formula. In our example, by using the quadratic formula, it allows for solving cases where other methods might not be applicable, especially when the roots are not conveniently rational or real numbers.

Coefficients

In the context of quadratic equations, coefficients are the constant multipliers of the variables present in the equation. For the equation \(ay^2 + by + c = 0\), the coefficients are:

• a: the coefficient of \(y^2\).
• b: the coefficient of \(y\).
• c: the constant term.

Recognizing and correctly identifying these coefficients is crucial in using the quadratic formula. For instance, in the exercise, \(a = 3\), \(b = -2\), and \(c = 4\) were identified as the coefficients. These values are then substituted into the quadratic formula to find the roots of the equation, making the determination of the coefficients a vital first step in solving quadratic equations.