x = -b +/- sqrt(b^2 - 4ac)2a
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve the quadratic equation $3x^2 + 7x - 15 = 0$, we use the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
In this equation, we have:
$a = 3$
$b = 7$
$c = -15$
Step 1: Substitute the values of $a$, $b$, and $c$ into the quadratic formula.
$$x = \frac{-(7) \pm \sqrt{(7)^2 - 4(3)(-15)}}{2(3)}$$
Step 2: Simplify the terms inside the square root and the denominator.
$$x = \frac{-7 \pm \sqrt{49 - (-180)}}{6}$$
$$x = \frac{-7 \pm \sqrt{49 + 180}}{6}$$
$$x = \frac{-7 \pm \sqrt{229}}{6}$$
Step 3: Calculate the two possible values for $x$.
$$x_1 = \frac{-7 + \sqrt{229}}{6}$$
$$x_2 = \frac{-7 - \sqrt{229}}{6}$$
The exact solutions are:
$$x_1 = \frac{-7 + \sqrt{229}}{6}$$
$$x_2 = \frac{-7 - \sqrt{229}}{6}$$
The final answers are $\boxed{x = \frac{-7 \pm \sqrt{229}}{6}}$.
