- Identify the coefficients of the quadratic equation \(4x^2 + 12x + 9 = 0\) as \(a = 4\), \(b = 12\), and \(c = 9\).
- Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Substitute the identified values to get \(x = \frac{-12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\).
- Compare with the given options to conclude that the correct choice is \(\boxed{b}\).
Explanation
- Identifying Coefficients
The given equation is \(4x^2 + 12x + 9 = 0\). This is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. By comparing the given equation to the standard form, we identify the coefficients as:
\(a = 4\)
\(b = 12\)
\(c = 9\)
- Recalling the Quadratic Formula
The quadratic formula, which provides the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\), is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is derived from the process of completing the square for a general quadratic equation.
- Substituting Coefficients
Now, we substitute the coefficients \(a = 4\), \(b = 12\), and \(c = 9\) into the quadratic formula:
- Replace \(b\) with \(12\), so \(-b\) becomes \(-12\).
- Replace \(b^2\) with \((12)^2\).
- Replace \(4ac\) with \(4 \times 4 \times 9\).
- Replace \(2a\) with \(2 \times 4\).
Substituting these into the formula, we get:
\[x = \frac{-12 \pm \sqrt{(12)^2 - 4(4)(9)}}{2(4)}\]
- Selecting the Correct Option
Comparing our derived expression, \[x = \frac{-12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\], with the provided options:
A\(x = \frac{12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\)
B\(x = \frac{-12 \pm \sqrt{12^2 - 4(4)(9)}}{2(4)}\)
C\(x = \frac{-12 \pm \sqrt{12^2 + 4(4)(9)}}{2(4)}\)
We can see that option b matches our result perfectly. Thus, the correct option is b.
Examples
The quadratic formula is a fundamental tool for solving equations that model various real-world phenomena. For example, if you are throwing a ball and its height at time \(t\) is given by a quadratic function, you can use the quadratic formula to determine the exact time \(t\) when the ball hits the ground (height = 0). Understanding how to correctly substitute coefficients into this formula is essential for finding these critical points in physics, engineering, and economics.