Sure, I'd be glad to help you solve this quadratic equation.
Firstly, the quadratic equation is given as 9x^2 - 7x - 5 = 0
The general form of a quadratic equation is ax^2 + bx + c = 0 where a, b and c are real numbers and 'a' must not be zero.
Here, a = 9, b = -7 and c = -5
In this problem, we are asked to find the roots of the given equation.
Let's start with the quadratic formula, which is given as x = (-b ± sqrt(b^2 - 4ac)) / (2a).
We can start by calculating the discriminant of the quadratic equation.
The discriminant in the quadratic formula is the part under the square root, i.e., \(b^2 - 4ac\).
Substitute the values of a, b and c into the formula, we get discriminant = (-7)^2 - 4 * 9 * (-5) = 49 + 180 = 229.
Since this value is greater than zero, it indicates that we have two real roots.
Next comes the calculation for the roots of the equation.
Substitute the values of a, b and the discriminant into the formula to get the roots, as follows:
root1 = [-(-7) + sqrt(229)] / (2 * 9) = 1.23 (after rounding)
root2 = [-(-7) - sqrt(229)] / (2 * 9) = -0.45 (after rounding)
Hence, the equation 9x^2 - 7x - 5 = 0 has two real roots which are x = 1.23 and x = -0.45.