To solve the equation [tex]\(8x^2 = 6 + 22x\)[/tex], we follow these steps:
1. Rearrange the equation to standard quadratic form:
[tex]\[8x^2 - 22x - 6 = 0\][/tex]
2. Use the quadratic formula to find the solutions:
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
Here, [tex]\(a = 8\)[/tex], [tex]\(b = -22\)[/tex], and [tex]\(c = -6\)[/tex].
3. Substitute these coefficients into the quadratic formula:
[tex]\[x = \frac{-(-22) \pm \sqrt{(-22)^2 - 4 \cdot 8 \cdot (-6)}}{2 \cdot 8}\][/tex]
[tex]\[x = \frac{22 \pm \sqrt{484 + 192}}{16}\][/tex]
[tex]\[x = \frac{22 \pm \sqrt{676}}{16}\][/tex]
[tex]\[x = \frac{22 \pm 26}{16}\][/tex]
4. Simplify to get the two solutions:
[tex]\[x = \frac{22 + 26}{16} = \frac{48}{16} = 3\][/tex]
[tex]\[x = \frac{22 - 26}{16} = \frac{-4}{16} = -\frac{1}{4}\][/tex]
Thus, the solutions to the equation [tex]\(8x^2 = 6 + 22x\)[/tex] are [tex]\(x = -\frac{1}{4}\)[/tex] and [tex]\(x = 3\)[/tex].
Checking the given values:
- [tex]\(x = -3\)[/tex]:
[tex]\[
8(-3)^2 = 6 + 22(-3) \][/tex]
[tex]\[72 \neq -60 \Rightarrow x = -3 \text{ is not a solution.}
\][/tex]
- [tex]\(x = -\frac{1}{4}\)[/tex]:
[tex]\[
8\left(-\frac{1}{4}\right)^2 = 6 + 22\left(-\frac{1}{4}\right) \][/tex]
[tex]\[
8\left(\frac{1}{16}\right) = 6 - \frac{22}{4} \][/tex]
[tex]\[0.5 = 0.5 \Rightarrow x = -\frac{1}{4} \text{ is a solution.}
\][/tex]
- [tex]\(x = 3\)[/tex]:
[tex]\[
8(3)^2 = 6 + 22(3) \][/tex]
[tex]\[
72 = 72 \Rightarrow x = 3 \text{ is a solution.}
\][/tex]
- [tex]\(x = 6\)[/tex]:
[tex]\[
8(6)^2 = 6 + 22(6) \][/tex]
[tex]\[
288 \neq 138 \Rightarrow x = 6 \text{ is not a solution.}
\][/tex]
Conclusion:
The correct solutions are:
[tex]\[x = -\frac{1}{4} \][/tex]
[tex]\[x = 3 \][/tex]
Thus, checking the boxes yields:
\- [tex]\(x = -3\)[/tex] [Not a solution]
- [tex]\(x = -\frac{1}{4}\)[/tex] [Solution]
- [tex]\(x = 3\)[/tex] [Solution]
- [tex]\(x = 6\)[/tex] [Not a solution]
