Certainly! Let's solve the given quadratic equation step-by-step by completing the square.
Given equation:
[tex]\[ 0 = -2x^2 + 2x + 3 \][/tex]
We start by moving the constant to the other side:
[tex]\[ -3 = -2x^2 + 2x \][/tex]
This is Step 1 that transforms the equation into:
[tex]\[ -3 = -2(x^2 - x) \][/tex]
Next, we complete the square inside the parentheses:
Starting with:
[tex]\[ x^2 - x \][/tex]
Complete the square by adding and subtracting [tex]\(\left(\frac{1}{2}\right)^2\)[/tex]:
[tex]\[ x^2 - x + \frac{1}{4} - \frac{1}{4} = (x - \frac{1}{2})^2 - \frac{1}{4} \][/tex]
Insert this back into the equation:
[tex]\[ -3 = -2((x - \frac{1}{2})^2 - \frac{1}{4}) \][/tex]
Then distribute [tex]\(-2\)[/tex] on the right-hand side:
[tex]\[ -3 = -2(x - \frac{1}{2})^2 + \frac{1}{2} \][/tex]
Now, move [tex]\(\frac{1}{2}\)[/tex] to the left side:
[tex]\[ -3 - \frac{1}{2} = -2(x - \frac{1}{2})^2 \][/tex]
Combine the terms on the left side:
[tex]\[ -\frac{7}{2} = -2(x - \frac{1}{2})^2 \][/tex]
Next, divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ \frac{7}{4} = (x - \frac{1}{2})^2 \][/tex]
Now, take the square root of both sides:
[tex]\[ \sqrt{\frac{7}{4}} = x - \frac{1}{2} \][/tex]
Simplify the square root:
[tex]\[ \frac{\sqrt{7}}{2} = x - \frac{1}{2} \][/tex]
Finally, isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{1}{2} \pm \frac{\sqrt{7}}{2} \][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[ x = \frac{1}{2} + \frac{\sqrt{7}}{2} \][/tex]
[tex]\[ x = \frac{1}{2} - \frac{\sqrt{7}}{2} \][/tex]
