We are tasked with solving the quadratic equation $ 2x^2 - 7x + 3 = 0 $ using the method of
completing the square. Let's go through the solution step by step.
---
Step 1: Start with the given equation
The given equation is:
$$
2x^2 - 7x + 3 = 0
$$
---
Step 2: Divide the entire equation by the coefficient of \( x^2 \)
To simplify the process, we divide every term in the equation by the coefficient of \( x^2 \), which is 2:
$$
\frac{2x^2}{2} - \frac{7x}{2} + \frac{3}{2} = 0
$$
This simplifies to:
$$
x^2 - \frac{7}{2}x + \frac{3}{2} = 0
$$
---
Step 3: Isolate the constant term
Move the constant term (\( \frac{3}{2} \)) to the right-hand side of the equation:
$$
x^2 - \frac{7}{2}x = -\frac{3}{2}
$$
---
Step 4: Complete the square
To complete the square, we need to add and subtract a specific value on the left-hand side. The value to add is determined by taking half of the coefficient of \( x \), squaring it, and then adding it to both sides.
- The coefficient of \( x \) is \( -\frac{7}{2} \).
- Half of this coefficient is:
$$
\frac{-\frac{7}{2}}{2} = -\frac{7}{4}
$$
- Square this value:
$$
\left( -\frac{7}{4} \right)^2 = \frac{49}{16}
$$
Now, add \( \frac{49}{16} \) to both sides of the equation:
$$
x^2 - \frac{7}{2}x + \frac{49}{16} = -\frac{3}{2} + \frac{49}{16}
$$
---
Step 5: Simplify the right-hand side
First, convert \( -\frac{3}{2} \) to a fraction with a denominator of 16:
$$
-\frac{3}{2} = -\frac{3 \cdot 8}{2 \cdot 8} = -\frac{24}{16}
$$
Now, add \( -\frac{24}{16} \) and \( \frac{49}{16} \):
$$
-\frac{24}{16} + \frac{49}{16} = \frac{-24 + 49}{16} = \frac{25}{16}
$$
So the equation becomes:
$$
x^2 - \frac{7}{2}x + \frac{49}{16} = \frac{25}{16}
$$
---
Step 6: Write the left-hand side as a perfect square
The left-hand side is now a perfect square trinomial:
$$
\left( x - \frac{7}{4} \right)^2 = \frac{25}{16}
$$
---
Step 7: Solve for \( x \)
Take the square root of both sides:
$$
x - \frac{7}{4} = \pm \sqrt{\frac{25}{16}}
$$
Since \( \sqrt{\frac{25}{16}} = \frac{5}{4} \), we have:
$$
x - \frac{7}{4} = \pm \frac{5}{4}
$$
This gives us two cases:
1. \( x - \frac{7}{4} = \frac{5}{4} \)
2. \( x - \frac{7}{4} = -\frac{5}{4} \)
Solve each case separately:
#### Case 1:
$$
x - \frac{7}{4} = \frac{5}{4}
$$
Add \( \frac{7}{4} \) to both sides:
$$
x = \frac{5}{4} + \frac{7}{4} = \frac{12}{4} = 3
$$
#### Case 2:
$$
x - \frac{7}{4} = -\frac{5}{4}
$$
Add \( \frac{7}{4} \) to both sides:
$$
x = -\frac{5}{4} + \frac{7}{4} = \frac{2}{4} = \frac{1}{2}
$$
---
Step 8: Write the final solutions
The solutions to the equation are:
$$
x = 3 \quad \text{and} \quad x = \frac{1}{2}
$$
---
Final Answer:
$$
\boxed{x = 3 \text{ or } x = \frac{1}{2}}
$$
Parent Tip: Review the logic above to help your child master the concept of completing the square and quadratic formula worksheet.
