Question

solve the quadratic equation.
2x + 2x² = 5
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

Rewrite the equation \(2x + 2x^{2}=5\) as \(2x^{2}+2x - 5 = 0\). Here, \(a = 2\), \(b = 2\), \(c=- 5\) for the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step2: Calculate discriminant

Compute the discriminant \(\Delta=b^{2}-4ac=(2)^{2}-4\times2\times(-5)=4 + 40 = 44\).

Step3: Apply quadratic formula

Substitute \(a\), \(b\), \(\Delta\) into the formula: \(x=\frac{-2\pm\sqrt{44}}{2\times2}=\frac{-2\pm2\sqrt{11}}{4}=\frac{-1\pm\sqrt{11}}{2}\).

Answer:

\(\frac{-1 + \sqrt{11}}{2}\), \(\frac{-1 - \sqrt{11}}{2}\)