91Ó°ÊÓ

The quadratic formula is used to find the solutions (or roots) of a quadratic equation. The standard form of a quadratic equation is given by:
ax^2 + bx + c = 0.
To find the roots of this equation, you can use the formula:
x = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
This formula will give you the values of x that satisfy the equation. In some cases, these values may be real numbers, while in others they might be complex numbers.
Always identify your coefficients a, b, and c before applying the formula.
In our example, a = 1, b = -6, and c = -91.
By substituting these values into the formula, we can find the roots of the equation.

The discriminant is a key part of the quadratic formula.
It is found under the square root sign in the formula:
b^2 - 4ac.
The value of the discriminant can tell us a lot about the nature of the roots of the quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.


• If it is zero, the equation has exactly one real root (the roots are repeated).


• If it is negative, the equation has two complex roots.

In our problem, we calculated the discriminant to be 400. Since 400 is a positive number, we have two distinct real roots.

In some quadratic equations, the discriminant can be negative, leading to complex roots.
Complex numbers are numbers that have both a real part and an imaginary part.
The imaginary unit is denoted by i, and it is defined by the property that i^2 = -1.
If you encounter a negative discriminant, you’ll need to use i to express the roots.
For example, if the discriminant were -400, instead of +400, the square root of -400 would be written as 20i.
Thus, the roots would be complex numbers.
In general, complex roots occur in conjugate pairs:
a + bi and a - bi.
For our solved problem, the roots were real, but knowing how to handle complex roots is essential for more advanced quadratic equations.

The roots of a quadratic equation are the values of x that satisfy the equation.
In the given exercise, we found the roots by applying the quadratic formula and solving step by step.
Here's a recap of how it went:

• We brought 100 to the other side of the equation to set it to zero: \(x^2 - 6x - 91 = 0 \).


• We identified the coefficients: a = 1, b = -6, c = -91.


• We used the quadratic formula to find the roots: \(x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-91)}}{2(1)}\).


• We calculated the discriminant: 400.


• We substituted the values and simplified to find the roots: 13 and -7.

Thus, the roots of the equation \(x^2 - 6x + 9 = 100 \) are x = 13 and x = -7.