- Convert the given equation \(3x^2 + 8x = 12\) into the standard form \(3x^2 + 8x - 12 = 0\).
- Identify coefficients \(a=3, b=8, c=-12\) and apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Calculate the discriminant \(D = 8^2 - 4(3)(-12) = 208\) and simplify the square root to \(4\sqrt{13}\).
- Obtain the roots \(x = \frac{-8 \pm 4\sqrt{13}}{6} = \frac{-4 \pm 2\sqrt{13}}{3}\), which corresponds to option \(\boxed{a}\).
Explanation
- Rewriting the Equation in Standard Form
The given equation is \(3x^2 + 8x = 12\). To find the roots, we first rewrite this equation in the standard quadratic form \(ax^2 + bx + c = 0\). Subtracting \(12\) from both sides, we get:
\[3x^2 + 8x - 12 = 0.\]
Here, the coefficients are \(a = 3\), \(b = 8\), and \(c = -12\).
- Applying the Quadratic Formula
To find the roots, we use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). First, we calculate the discriminant, which is given by \(b^2 - 4ac\):
\[b^2 - 4ac = 8^2 - 4(3)(-12) = 64 + 144 = 208.\]
The discriminant is 208. Now we substitute these values into the quadratic formula:
\[x = \frac{-8 \pm \sqrt{208}}{2(3)}.\]
- Simplifying the Results and Selecting the Answer
We can simplify the expression \(\sqrt{208}\). Since \(208 = 16 \times 13\), we have \(\sqrt{208} = \sqrt{16} \times \sqrt{13} = 4\sqrt{13}\). Substituting this back into our expression for \(x\), we get:
\[x = \frac{-8 \pm 4\sqrt{13}}{6}.\]
Dividing the numerator and the denominator by 2:
\[x = \frac{-4 \pm 2\sqrt{13}}{3}.\]
This matches option (a).
Examples
Quadratic equations are fundamental in physics and engineering. For example, when you throw a ball, its height over time follows a quadratic path. Solving equations like these allows you to determine exactly when the ball reaches a certain height or when it hits the ground. This specific skill is essential for predicting trajectories and optimizing designs in various technical fields.