Question

solve the quadratic equation.
3x² + 4x = 1
write one exact solution in each box. you can add or remove boxes. if there are no real solutions, remove all boxes.

Explanation:

Step1: Rewrite in standard form

First, rewrite the quadratic equation \(3x^2 + 4x = 1\) in standard form \(ax^2+bx+c = 0\) by subtracting 1 from both sides:
\(3x^2 + 4x - 1=0\)
Here, \(a = 3\), \(b = 4\), and \(c=- 1\).

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Substitute \(a = 3\), \(b = 4\), and \(c=-1\) into the formula:
First, calculate the discriminant \(\Delta=b^2-4ac=(4)^2-4\times3\times(-1)=16 + 12=28\)
Then, \(x=\frac{-4\pm\sqrt{28}}{2\times3}=\frac{-4\pm2\sqrt{7}}{6}=\frac{-2\pm\sqrt{7}}{3}\)

Answer:

\(\frac{-2 + \sqrt{7}}{3}\), \(\frac{-2-\sqrt{7}}{3}\)