Questions: Solve the equation using any method. 3x^2 + 4 = 13x

Solution Steps

To solve the quadratic equation \(3x^2 + 4 = 13x\), we first rearrange it into the standard form \(ax^2 + bx + c = 0\). Then, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).

The given equation is \(3x^2 + 4 = 13x\). We rearrange it into the standard quadratic form: \[ 3x^2 - 13x + 4 = 0 \]

Identify the coefficients from the quadratic equation \(ax^2 + bx + c = 0\):

• \(a = 3\)
• \(b = -13\)
• \(c = 4\)

The discriminant \(\Delta\) is calculated as: \[ \Delta = b^2 - 4ac = (-13)^2 - 4 \times 3 \times 4 = 169 - 48 = 121 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\), we find the solutions: \[ x_1 = \frac{-(-13) + \sqrt{121}}{2 \times 3} = \frac{13 + 11}{6} = \frac{24}{6} = 4 \] \[ x_2 = \frac{-(-13) - \sqrt{121}}{2 \times 3} = \frac{13 - 11}{6} = \frac{2}{6} = \frac{1}{3} \]

Final Answer