Problems Based on Formula for Roots

Quadratic equations are polynomial equations of degree two. The general form of a quadratic equation is:

$$ ax^2 + bx + c = 0 $$

where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ).

Formula for Roots

The roots of a quadratic equation can be found using the quadratic formula:

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

Here, ( x ) represents the roots of the quadratic equation, and the term under the square root, ( b^2 - 4ac ), is known as the discriminant, denoted by ( D ).

Discriminant and Nature of Roots

The discriminant determines the nature of the roots of the quadratic equation:

Discriminant (D)	Nature of Roots	Description
( D > 0 )	Real and Distinct	Two different real roots
( D = 0 )	Real and Equal	Two identical real roots
( D < 0 )	Complex	Two complex roots (conjugate pair)

Examples

Example 1: Real and Distinct Roots

Consider the quadratic equation ( 2x^2 - 4x + 1 = 0 ).

1. Calculate the discriminant:

$$ D = b^2 - 4ac = (-4)^2 - 4(2)(1) = 16 - 8 = 8 $$

Since ( D > 0 ), the roots are real and distinct.

1. Find the roots using the quadratic formula:

$$ x = \frac{-(-4) \pm \sqrt{8}}{2(2)} = \frac{4 \pm 2\sqrt{2}}{4} $$

So, the roots are ( x = 1 + \frac{\sqrt{2}}{2} ) and ( x = 1 - \frac{\sqrt{2}}{2} ).

Example 2: Real and Equal Roots

Consider the quadratic equation ( x^2 - 6x + 9 = 0 ).

1. Calculate the discriminant:

$$ D = b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0 $$

Since ( D = 0 ), the roots are real and equal.

1. Find the roots using the quadratic formula:

$$ x = \frac{-(-6) \pm \sqrt{0}}{2(1)} = \frac{6}{2} = 3 $$

So, both roots are ( x = 3 ).

Example 3: Complex Roots

Consider the quadratic equation ( x^2 + 4x + 8 = 0 ).

1. Calculate the discriminant:

$$ D = b^2 - 4ac = (4)^2 - 4(1)(8) = 16 - 32 = -16 $$

Since ( D < 0 ), the roots are complex.

1. Find the roots using the quadratic formula:

$$ x = \frac{-4 \pm \sqrt{-16}}{2(1)} = \frac{-4 \pm 4i}{2} = -2 \pm 2i $$

So, the roots are ( x = -2 + 2i ) and ( x = -2 - 2i ).

Important Points to Remember

• The quadratic formula can always be used to find the roots of any quadratic equation, regardless of the discriminant's value.
• The discriminant provides crucial information about the nature of the roots without actually calculating them.
• When solving problems, it's essential to simplify the roots as much as possible, especially when dealing with irrational or complex numbers.
• For complex roots, they always occur in conjugate pairs, meaning if ( a + bi ) is a root, then ( a - bi ) is also a root.

Practice Problems

1. Find the roots of the quadratic equation ( 3x^2 - 2x - 5 = 0 ).
2. Determine the nature of the roots of ( x^2 - 4x + 13 = 0 ) without finding the actual roots.
3. Solve ( 5x^2 + 3x + 1 = 0 ) and express the roots in simplest radical form.

By understanding the formula for roots and the discriminant, students can effectively solve quadratic equations and predict the nature of their roots, which is a fundamental skill in algebra and is widely applicable in various fields of mathematics and science.