Question

solve the quadratic equation.
7x² + 2 = -x
write one exact, simplified solution in each box. you can add or remove boxes. if a solution is not a real number, write it in the form a + bi or a - bi, where a and b are real numbers.

Explanation:

Step1: Rewrite in standard form

First, we rewrite the quadratic equation \(7x^{2}+2 = -x\) in standard form \(ax^{2}+bx + c = 0\). We add \(x\) to both sides to get \(7x^{2}+x + 2=0\). Here, \(a = 7\), \(b = 1\), and \(c = 2\).

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 7\), \(b = 1\), and \(c = 2\) into the formula. First, calculate the discriminant \(\Delta=b^{2}-4ac=(1)^{2}-4\times7\times2=1 - 56=- 55\). Then, \(x=\frac{-1\pm\sqrt{-55}}{2\times7}=\frac{-1\pm i\sqrt{55}}{14}\) (since \(\sqrt{-55}=i\sqrt{55}\) where \(i\) is the imaginary unit).

Answer:

\(\frac{-1 + i\sqrt{55}}{14}\), \(\frac{-1 - i\sqrt{55}}{14}\)