Question

Solve the equation. $$3 x^{2}+11 x+10=0$$

Short answer

The solutions for the equation \(3x^{2}+11x+10=0\) are \(x_{1}=-5/3\) and \(x_{2}=-2\).

Step-by-step solution

- Identify the coefficients

In the quadratic equation, the general form is \(ax^{2}+bx+c=0\). Therefore, comparing our given equation with general form, we can safely say that \(a=3\), \(b=11\), and \(c=10\).

- Apply the quadratic formula

Now we use the quadratic formula, which is \[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\] We substitute our coefficients values (\(a=3\), \(b=11\), \(c=10\)) into the formula. This will then become: \[x=\frac{-11\pm\sqrt{11^{2}-4(3)(10)}}{2(3)}\] simplifying under the square root we get: \[x=\frac{-11\pm\sqrt{121-120}}{6}\]

- Solve for \(x\)

Continuing the calculation of \(x\) from step 2, we get: \[x=\frac{-11\pm\sqrt{1}}{6}\] This will yield two solutions, one for the plus and one for the minus: \(x_{1}=\frac{-11+1}{6}=-5/3\) and \(x_{2}=\frac{-11-1}{6}=-2\)