Solve the quadratic equation 4y^2 +5y-21 using completing the square method

To solve the quadratic equation \(4y^2 + 5y - 21 = 0\) using the method of completing the square, follow these steps:

1. **Divide through by the coefficient of \(y^2\)**:

Since the coefficient of \(y^2\) is 4, divide the entire equation by 4:
\[
y^2 + \frac{5}{4}y - \frac{21}{4} = 0
\]

2. **Move the constant term to the other side**:

\[
y^2 + \frac{5}{4}y = \frac{21}{4}
\]

3. **Complete the square**:

To complete the square, take the coefficient of \(y\) (which is \(\frac{5}{4}\)), divide it by 2, and square it:
\[
\left(\frac{\frac{5}{4}}{2}\right)^2 = \left(\frac{5}{8}\right)^2 = \frac{25}{64}
\]

Add this value to both sides of the equation:
\[
y^2 + \frac{5}{4}y + \frac{25}{64} = \frac{21}{4} + \frac{25}{64}
\]

4. **Simplify the right side**:

To add the fractions on the right side, find a common denominator. The least common denominator between 4 and 64 is 64. Hence:
\[
\frac{21}{4} = \frac{21 \times 16}{4 \times 16} = \frac{336}{64}
\]

So,
\[
\frac{336}{64} + \frac{25}{64} = \frac{361}{64}
\]

Therefore, the equation is:
\[
y^2 + \frac{5}{4}y + \frac{25}{64} = \frac{361}{64}
\]

5. **Write the left side as a perfect square**:

The left side of the equation can be written as the square of a binomial:
\[
\left(y + \frac{5}{8}\right)^2 = \frac{361}{64}
\]

6. **Take the square root of both sides**:

Take the square root of both sides, remembering to consider both the positive and negative roots:
\[
y + \frac{5}{8} = \pm \sqrt{\frac{361}{64}}
\]

\[
y + \frac{5}{8} = \pm \frac{19}{8}
\]

7. **Solve for \(y\)**:

Now, solve for \(y\) by isolating it in each case:
\[
y = -\frac{5}{8} + \frac{19}{8} \quad \text{or} \quad y = -\frac{5}{8} - \frac{19}{8}
\]

Simplify both solutions:
\[
y = \frac{14}{8} = \frac{7}{4} \quad \text{or} \quad y = -\frac{24}{8} = -3
\]

So the solutions to the equation \(4y^2 + 5y - 21 = 0\) are:
\[
y = \frac{7}{4} \quad \text{and} \quad y = -3
\]