Sure, let's solve each quadratic equation step-by-step:
### Equation 1: [tex]\(2x^2 - 3 = 29\)[/tex]
1. Add 3 to both sides:
[tex]\[
2x^2 - 3 + 3 = 29 + 3 \implies 2x^2 = 32
\][/tex]
2. Divide by 2:
[tex]\[
x^2 = \frac{32}{2} \implies x^2 = 16
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm \sqrt{16} \implies x = \pm 4
\][/tex]
So, the solution is [tex]\(x = 4, -4\)[/tex], which matches with (b).
### Equation 2: [tex]\(-1 - 5x^2 = -321\)[/tex]
1. Add 1 to both sides:
[tex]\[
-1 + 1 - 5x^2 = -321 + 1 \implies -5x^2 = -320
\][/tex]
2. Divide by -5:
[tex]\[
x^2 = \frac{-320}{-5} \implies x^2 = 64
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm \sqrt{64} \implies x = \pm 8
\][/tex]
So, the solution is [tex]\(x = 8, -8\)[/tex], which matches with (c).
### Equation 3: [tex]\(3x^2 = 75\)[/tex]
1. Divide by 3:
[tex]\[
x^2 = \frac{75}{3} \implies x^2 = 25
\][/tex]
2. Take the square root of both sides:
[tex]\[
x = \pm \sqrt{25} \implies x = \pm 5
\][/tex]
So, the solution is [tex]\(x = 5, -5\)[/tex], which matches with (e).
### Equation 4: [tex]\(8x^2 + 9 = 313\)[/tex]
1. Subtract 9 from both sides:
[tex]\[
8x^2 + 9 - 9 = 313 - 9 \implies 8x^2 = 304
\][/tex]
2. Divide by 8:
[tex]\[
x^2 = \frac{304}{8} \implies x^2 = 38
\][/tex]
3. Take the square root of both sides (and round to the nearest tenth):
[tex]\[
x = \pm \sqrt{38} \approx \pm 6.2
\][/tex]
So, the solution is [tex]\(x \approx 6.2, -6.2\)[/tex], which matches with (a).
### Equation 5: [tex]\(32 = 25x^2 - 4\)[/tex]
1. Add 4 to both sides:
[tex]\[
32 + 4 = 25x^2 - 4 + 4 \implies 36 = 25x^2
\][/tex]
2. Divide by 25:
[tex]\[
x^2 = \frac{36}{25} \implies x^2 = 1.44
\][/tex]
3. Take the square root of both sides (and round to the nearest tenth):
[tex]\[
x = \pm \sqrt{1.44} \approx \pm 1.2
\][/tex]
So, the solution is [tex]\(x \approx 1.2, -1.2\)[/tex], which matches with (d).
### Summary:
1. [tex]\(2x^2 - 3 = 29\)[/tex] matches with (b): [tex]\(x=4, -4\)[/tex]
2. [tex]\(-1 - 5x^2 = -321\)[/tex] matches with (c): [tex]\(x=8, -8\)[/tex]
3. [tex]\(3x^2 = 75\)[/tex] matches with (e): [tex]\(x=5, -5\)[/tex]
4. [tex]\(8x^2 + 9 = 313\)[/tex] matches with (a): [tex]\(x=6.2, -6.2\)[/tex]
5. [tex]\(32 = 25x^2 - 4\)[/tex] matches with (d): [tex]\(x=1.2, -1.2\)[/tex]
