Solve by the quadratic formula

Introduction to the Quadratic Formula

The quadratic formula is an important and versatile tool for solving quadratic equations.
Quadratic equations are of the form ax^2 + bx + c = 0, where a, b, and c are coefficients and x is the unknown variable.
The quadratic formula is illustrated as: x = (-b ± sqrt(b^2 - 4ac)) / (2a).

Solving Quadratic Equations by the Quadratic Formula

The main advantage of the quadratic formula is that it can solve any quadratic equation, even if it could not be easily factored or simplified by other methods.
To find the roots of the quadratic equation using the quadratic formula, carefully input values for a, b, and c into the formula.
It’s important to note the “±” symbol in the formula, which means there can be two possible solutions: one using plus (known as the positive root) and one using minus (known as the negative root).
When b^2 - 4ac is positive, the quadratic equation will have two real roots.
When b^2 - 4ac equals zero, the equation has exactly one real root.
When b^2 - 4ac is negative, the equation has two complex roots, and you’ll need knowledge of imaginary numbers to understand these solutions.

Examples

For example, let’s use the quadratic equation 3x^2 + 2x - 1 = 0. In this case, a = 3, b = 2, and c = -1
Substitute these values into the quadratic formula: x =(-2 ± sqrt((2)^2 - 43(-1))) / (2*3)
Simplify the square root: x =(-2 ± sqrt(4 +12)) / 6
This leads to x =(-2 ± sqrt(16)) / 6, after simplification, x = -2/3 or 1/2

Conclusion

The quadratic formula is very reliable because you can use it to solve any quadratic equation, but it should be used with caution due to potential complexity.
The discriminant (b^2 - 4ac) determines the nature of the roots such as real, repeated or complex.
Practise is key to mastering this topic, and a solid understanding of the quadratic formula can support your progress in advanced mathematical topics.