To solve the quadratic equation [tex]\(9x^2 - 29x - 28 = 0\)[/tex], we need to use the quadratic formula. The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\(a = 9\)[/tex]
- [tex]\(b = -29\)[/tex]
- [tex]\(c = -28\)[/tex]
### Step 1: Calculate the Discriminant
The discriminant [tex]\(D\)[/tex] is calculated using the formula [tex]\(D = b^2 - 4ac\)[/tex]:
[tex]\[
D = (-29)^2 - 4 \cdot 9 \cdot (-28)
\][/tex]
First, calculate [tex]\((-29)^2\)[/tex]:
[tex]\[
(-29)^2 = 841
\][/tex]
Then calculate [tex]\(4 \cdot 9 \cdot (-28)\)[/tex]:
[tex]\[
4 \cdot 9 \cdot (-28) = -1008
\][/tex]
Now, compute the discriminant:
[tex]\[
D = 841 - (-1008) = 841 + 1008 = 1849
\][/tex]
### Step 2: Calculate the Roots
Using the quadratic formula, we find the two roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x_1 = \frac{{-b + \sqrt{D}}}{2a} \quad \text{and} \quad x_2 = \frac{{-b - \sqrt{D}}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(D\)[/tex]:
[tex]\[
x_1 = \frac{{-(-29) + \sqrt{1849}}}{2 \cdot 9} \quad \text{and} \quad x_2 = \frac{{-(-29) - \sqrt{1849}}}{2 \cdot 9}
\][/tex]
Since [tex]\(-(-29) = 29\)[/tex]:
[tex]\[
x_1 = \frac{{29 + \sqrt{1849}}}{18} \quad \text{and} \quad x_2 = \frac{{29 - \sqrt{1849}}}{18}
\][/tex]
Calculate [tex]\(\sqrt{1849}\)[/tex]:
[tex]\[
\sqrt{1849} = 43
\][/tex]
Now, substitute back:
[tex]\[
x_1 = \frac{{29 + 43}}{18} = \frac{{72}}{18} = 4.0
\][/tex]
[tex]\[
x_2 = \frac{{29 - 43}}{18} = \frac{{-14}}{18} = -0.78
\][/tex]
Thus, the solutions to the equation [tex]\(9x^2 - 29x - 28 = 0\)[/tex] are:
[tex]\[
\boxed{4.0 \quad \text{and} \quad -0.78}
\][/tex]
