solve quadratic equation by completing the square: y²-7y-12

To solve the quadratic equation \( y^2 - 7y - 12 = 0 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
Start by moving the constant term (\(-12\)) to the right side of the equation:

\[
y^2 - 7y = 12
\]

### Step 2: Complete the square
To complete the square, we need to add a term to both sides of the equation that makes the left side a perfect square trinomial.

Take the coefficient of the \( y \) term (\(-7\)), divide it by 2, and then square the result:

\[
\left(\frac{-7}{2}\right)^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4}
\]

Add \(\frac{49}{4}\) to both sides:

\[
y^2 - 7y + \frac{49}{4} = 12 + \frac{49}{4}
\]

### Step 3: Simplify both sides
Now, the left side of the equation is a perfect square trinomial:

\[
\left(y - \frac{7}{2}\right)^2 = 12 + \frac{49}{4}
\]

To simplify the right side, convert 12 to a fraction with a denominator of 4:

\[
12 = \frac{48}{4}
\]

So,

\[
\frac{48}{4} + \frac{49}{4} = \frac{97}{4}
\]

Thus, the equation becomes:

\[
\left(y - \frac{7}{2}\right)^2 = \frac{97}{4}
\]

### Step 4: Take the square root of both sides
Now, take the square root of both sides to solve for \( y \):

\[
y - \frac{7}{2} = \pm \sqrt{\frac{97}{4}}
\]

Simplify the square root:

\[
y - \frac{7}{2} = \pm \frac{\sqrt{97}}{2}
\]

### Step 5: Solve for \( y \)
Finally, add \(\frac{7}{2}\) to both sides:

\[
y = \frac{7}{2} \pm \frac{\sqrt{97}}{2}
\]

So, the solutions are:

\[
y = \frac{7 + \sqrt{97}}{2} \quad \text{or} \quad y = \frac{7 - \sqrt{97}}{2}
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you determine when to use completing the square versus other methods?
2. What are the advantages of solving quadratic equations by completing the square?
3. How does completing the square help in understanding the vertex form of a quadratic equation?
4. Can you derive the quadratic formula using the method of completing the square?
5. How can completing the square be used in solving real-world problems?

**Tip:** Completing the square is a valuable technique for rewriting quadratic equations in a form that reveals the vertex of the parabola, which is useful for graphing and solving equations.

Learn how to solve the quadratic equation y^2 - 7y - 12 by completing the square with a step-by-step solution. This problem covers key algebraic concepts, including completing the square and quadratic equations. Suitable for students in Grades 9-12.

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