Question - Solving Quadratic Equation using Quadratic Formula
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Solution:
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 $$3x^2 - 7x - 32 = 0$$.
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