Solving Quadratic Equation using Quadratic Formula
The equation provided in the image is a quadratic equation, which is in the form \(ax^2 + bx + c = 0\):
\(3x^2 - 7x - 32 = 0\)
To solve this quadratic equation, we can either factorise it, complete the square, or use the quadratic formula. The quadratic formula is:
\(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\)
Here, \(a = 3\), \(b = -7\), and \(c = -32\). Let's use the quadratic formula:
\(x = \frac{{-(-7) \pm \sqrt{{(-7)^2 - 4(3)(-32)}}}}{{2(3)}}\)
\(x = \frac{{7 \pm \sqrt{{49 + 384}}}}{{6}}\)
\(x = \frac{{7 \pm \sqrt{{433}}}}{{6}}\)
Since \(433\) is a prime number, it cannot be simplified further into a perfect square, so the solutions to the equation are:
\(x = \frac{{7 + \sqrt{433}}}{6}\) and \(x = \frac{{7 - \sqrt{433}}}{6}\)
These are the roots of the quadratic equation \(3x^2 - 7x - 32 = 0\).
