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:
\[
y = \frac{7}{4} \quad \text{and} \quad y = -3
\]
Solve the quadratic equation 4y^2 +5y-21 using completing the square method
1 answer