Sure, I'd be happy to help you solve the quadratic equation [tex]\(4b^2 + 8b + 7 = 4\)[/tex] using the quadratic formula step-by-step!
### 1. Simplify the Equation
First, we need to rewrite the equation in standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[4b^2 + 8b + 7 - 4 = 0 \][/tex]
This simplifies to:
[tex]\[4b^2 + 8b + 3 = 0 \][/tex]
### 2. Identify the Coefficients
From the simplified quadratic equation [tex]\(4b^2 + 8b + 3 = 0\)[/tex], identify the coefficients:
[tex]\[ a = 4 \][/tex]
[tex]\[ b = 8 \][/tex]
[tex]\[ c = 3 \][/tex]
### 3. Calculate the Discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the identified coefficients:
[tex]\[ \Delta = 8^2 - 4 \cdot 4 \cdot 3 \][/tex]
[tex]\[ \Delta = 64 - 48 \][/tex]
[tex]\[ \Delta = 16 \][/tex]
### 4. Check the Discriminant
The discriminant [tex]\(\Delta = 16\)[/tex] is greater than 0, which indicates that there are two distinct real roots.
### 5. Apply the Quadratic Formula
The quadratic formula is:
[tex]\[ b = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the discriminant and the coefficients into the quadratic formula:
[tex]\[ b = \frac{-8 \pm \sqrt{16}}{2 \cdot 4} \][/tex]
### 6. Simplify Further
First, solve for [tex]\(\sqrt{16}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
Now substitute back into the formula:
[tex]\[b = \frac{-8 \pm 4}{8} \][/tex]
This gives us two roots, one for the [tex]\(+\)[/tex] and one for the [tex]\(-\)[/tex]:
Root 1 (using the plus sign [tex]\(+\)[/tex]):
[tex]\[ b_1 = \frac{-8 + 4}{8} = \frac{-4}{8} = -0.5 \][/tex]
Root 2 (using the minus sign [tex]\(-\)[/tex]):
[tex]\[ b_2 = \frac{-8 - 4}{8} = \frac{-12}{8} = -1.5 \][/tex]
### 7. Conclusion
The discriminant is [tex]\(\Delta = 16\)[/tex], and the two roots of the equation [tex]\(4b^2 + 8b + 3 = 0\)[/tex] are:
[tex]\[ b_1 = -0.5 \][/tex]
[tex]\[ b_2 = -1.5 \][/tex]
So, the final solution is:
[tex]\[ \Delta = 16, \quad b_1 = -0.5, \quad b_2 = -1.5 \][/tex]
