Determining the Nature of Roots for Quadratic Equation 4x² - 2x - 3 = 0

To determine the nature of the roots of a quadratic equation in the standard form \(ax^2 + bx + c = 0\), we analyze the discriminant, which is given by the formula \(\Delta = b^2 - 4ac\).

The nature of the roots depends on the value of the discriminant (\(\Delta\)) as follows:

• If \(\Delta > 0\), the roots are real and unequal.
  • If \(\Delta\) is a perfect square, the roots are rational and unequal.
  • If \(\Delta\) is not a perfect square, the roots are irrational and unequal.
• If \(\Delta = 0\), the roots are real and equal (also rational).
• If \(\Delta < 0\), the roots are imaginary (or non-real) and unequal.

Calculating the Discriminant for 4x² - 2x - 3 = 0

The given quadratic equation is \(4x^2 - 2x - 3 = 0\).

Comparing this to the standard form \(ax^2 + bx + c = 0\), we can identify the coefficients:

• \(a = 4\)
• \(b = -2\)
• \(c = -3\)

Now, let's calculate the discriminant \(\Delta\):

\(\Delta = b^2 - 4ac\)

Substitute the values of \(a\), \(b\), and \(c\) into the formula:

\(\Delta = (-2)^2 - 4(4)(-3)\)

Calculate the terms:

\(\Delta = 4 - (16)(-3)\)

\(\Delta = 4 - (-48)\)

\(\Delta = 4 + 48\)

\(\Delta = 52\)

Interpreting the Discriminant Value (Δ = 52)

We found that the discriminant \(\Delta = 52\).

Let's analyze this value:

1. Is \(\Delta > 0\)? Yes, \(52 > 0\). This means the roots are real and unequal.
2. Since the roots are real and unequal, is \(\Delta\) a perfect square? A perfect square is an integer that is the square of another integer (e.g., 1, 4, 9, 16, 25, 36, 49, 64, ...). 52 is not a perfect square because \(\sqrt{52}\) is not an integer (\(\sqrt{49} = 7\) and \(\sqrt{64} = 8\)).

Since \(\Delta > 0\) and \(\Delta\) is not a perfect square, the roots are irrational and unequal.

Conclusion on Nature of Roots

Based on the analysis of the discriminant \(\Delta = 52\), the nature of the roots of the equation \(4x^2 - 2x - 3 = 0\) is irrational and unequal.

Revision Table: Discriminant and Nature of Roots

Additional Information: The Quadratic Formula

The roots of a quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula:

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

Notice that the term inside the square root is the discriminant, \(\Delta = b^2 - 4ac\). This formula directly shows how the discriminant affects the roots:

• If \(\Delta > 0\), \(\sqrt{\Delta}\) is a real number (either rational or irrational), leading to two distinct real roots: \(\frac{-b + \sqrt{\Delta}}{2a}\) and \(\frac{-b - \sqrt{\Delta}}{2a}\).
• If \(\Delta = 0\), \(\sqrt{\Delta} = 0\), leading to one real root (or two equal real roots): \(\frac{-b}{2a}\).
• If \(\Delta < 0\), \(\sqrt{\Delta}\) is an imaginary number (\(\sqrt{-k} = i\sqrt{k}\) for \(k > 0\)), leading to two distinct complex conjugate roots.

For the equation \(4x^2 - 2x - 3 = 0\), the roots are:

\(x = \frac{-(-2) \pm \sqrt{52}}{2(4)}\)

\(x = \frac{2 \pm \sqrt{52}}{8}\)

\(x = \frac{2 \pm \sqrt{4 \times 13}}{8}\)

\(x = \frac{2 \pm 2\sqrt{13}}{8}\)

\(x = \frac{2(1 \pm \sqrt{13})}{8}\)

\(x = \frac{1 \pm \sqrt{13}}{4}\)

The roots are \(\frac{1 + \sqrt{13}}{4}\) and \(\frac{1 - \sqrt{13}}{4}\). Since \(\sqrt{13}\) is irrational, these roots are irrational and they are clearly unequal.