To rewrite the equation \(2 x^{2} + 7 x + 6 = 0\) by completing the square, follow these steps:
- First, divide the entire equation by 2 to make the coefficient of \(x^2\) equal to 1:
\[ x^2 + \frac{7}{2} x + 3 = 0 \]
- Rearrange the equation to isolate the constant on the other side:
\[ x^2 + \frac{7}{2} x = -3 \]
- To complete the square, take half of the coefficient of \(x\) (which is \(\frac{7}{2}\)), square it, and add it to both sides. Half of \(\frac{7}{2}\) is \(\frac{7}{4}\), and squaring it gives \(\left(\frac{7}{4}\right)^2 = \frac{49}{16}\):
\[ x^2 + \frac{7}{2} x + \frac{49}{16} = -3 + \frac{49}{16} \]
- Convert -3 to a fraction with a denominator of 16:
\[ -3 = -\frac{48}{16} \]
- Now simplify the right side:
\[ -3 + \frac{49}{16} = -\frac{48}{16} + \frac{49}{16} = \frac{1}{16} \]
- Now, rewrite the left side as a squared binomial:
\[ \left(x + \frac{7}{4}\right)^2 = \frac{1}{16} \]
Putting it all together, the completed square form of the equation is:
\[ \left(x + \frac{7}{4}\right)^2 = \frac{1}{16} \]
So, the final answer is:
\[ (x + \frac{7}{4})^2 = \frac{1}{16} \]
1 answer