To solve the quadratic equation [tex]\(0 = 4x^2 + 12x + 9\)[/tex], we will follow these steps using the quadratic formula which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
1. Identify the coefficients: For the given quadratic equation [tex]\(4x^2 + 12x + 9 = 0\)[/tex], we recognize that:
[tex]\[ a = 4, \quad b = 12, \quad c = 9 \][/tex]
2. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 12^2 - 4 \cdot 4 \cdot 9 \][/tex]
[tex]\[ \Delta = 144 - 144 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
3. Evaluate the roots: Since the discriminant [tex]\(\Delta = 0\)[/tex], it means we have a perfect square and thus a single (repeated) real root. Substituting the values into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-12 \pm \sqrt{0}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{-12 \pm 0}{8} \][/tex]
[tex]\[ x = \frac{-12}{8} \][/tex]
[tex]\[ x = -1.5 \][/tex]
Since the discriminant is zero, there is only one unique root, but it is counted with multiplicity two. Therefore, the solution is that [tex]\(x\)[/tex] is [tex]\( -1.5 \)[/tex].
So, the correct equation that shows the substitution of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula is:
[tex]\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4} \][/tex]
This correctly reflects the steps to find the root of the quadratic equation [tex]\( 0 = 4x^2 + 12x + 9 \)[/tex].
