Question - Solving Quadratic Equations with Quadratic Formula
Solution:
The image contains a mathematical expression:\[ 3x^2 + x - 10 \]To solve this quadratic equation, we would typically first try to factor it, if possible. However, if the quadratic doesn't factor easily or at all, then we can use the quadratic formula to find the roots of the equation, which is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this case, the coefficients for our quadratic expression $$3x^2 + x - 10$$ are $$a = 3$$, $$b = 1$$, and $$c = -10$$. Let's apply the quadratic formula:\[ x = \frac{-1 \pm \sqrt{1^2 - 4(3)(-10)}}{2(3)} \]\[ x = \frac{-1 \pm \sqrt{1 + 120}}{6} \]\[ x = \frac{-1 \pm \sqrt{121}}{6} \]\[ x = \frac{-1 \pm 11}{6} \]This gives us two solutions:\[ x = \frac{-1 + 11}{6} = \frac{10}{6} = \frac{5}{3} \]\[ x = \frac{-1 - 11}{6} = -2 \]Therefore, the two solutions to the equation $$3x^2 + x - 10 = 0$$ are $$x = \frac{5}{3}$$ and $$x = -2$$.
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