Question

Find the nature of the roots of the equation \(4 \mathrm{x}^{2}-2 \mathrm{x}-1=0\). (1) real and equal (2) rational and unequal (3) irrational and unequal (4) imaginary

Short answer

Answer: The roots are irrational and unequal.

Step-by-step solution

Calculate the Discriminant

Using the discriminant formula, plug in the values of a, b, and c: $$D = (-2)^2 - 4(4)(-1)$$

Solve for the Discriminant

Calculate the value of D: $$D = 4 + 16 = 20$$

Determine the Nature of the Roots

Since \(D = 20 > 0\) and \(20\) is not a perfect square, the roots are irrational and unequal. Therefore, the correct answer is option (3).

Key concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It typically has the form:

• ax² + bx + c = 0

Here, a, b, and c are constants, with a not equal to zero. The reason a cannot be zero is because if it were zero, the equation would no longer be quadratic but linear.

The solution to a quadratic equation can have one of three possible forms: two distinct solutions, one repeated solution, or no real solution at all. This is determined using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Notice the under-the-root expression, b² - 4ac. This is known as the discriminant, and it plays a crucial role in determining the nature of the roots of a quadratic equation.

Discriminant

The discriminant of a quadratic equation is a specific expression that can tell us a lot about the roots of the equation. It is given by the formula: \[ D = b^2 - 4ac \] Here, the discriminant, D, indicates important information:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has exactly one real root, which is repeated.
• If D < 0, there are no real roots; instead, the equation has two complex roots.

In our exercise, the quadratic equation was evaluated at a D = 20. As 20 > 0 and since 20 is not a perfect square, it clearly reveals that the roots are both real and distinct, but they are not rational.

Irrational Roots

Irrational roots arise when the discriminant is positive but not a perfect square. These roots cannot be simplified to exact fractions or integers. They are expressed in radical (or square root) form, leading to numbers that can be approximated but not exactly expressed as a simple fraction.

For example, the discriminant in our scenario gave us a value of 20, which is not a perfect square. The presence of a square root of a non-perfect square under the quadratic formula's square root term means the roots are irrational. These types of roots will often appear as non-repeating, non-terminating decimals when computed using a calculator.

Unequal Roots

Unequal roots occur in a quadratic equation when the discriminant is positive, indicating two different solutions. They are also known as distinct roots, meaning the two solutions you find are not equal to each other.

In our specific exercise, the discriminant was calculated to be greater than zero, proving that the roots are unequal. Since the discriminant was not a perfect square, these roots were also confirmed to be irrational, adding another layer to their distinction.

This means when you solve the equation using the quadratic formula, you'll end up with two distinct values for x, each yielding a different solution to the equation, exemplifying what unequal roots stand for.