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Let's solve each quadratic equation step by step and match them to their corresponding answers. We will round the solutions to the nearest tenth where necessary.
### Quadratic Equation 1: [tex]\(-1 - 5x^2 = -321\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ -1 - 5x^2 = -321 \implies -5x^2 = -320 \implies 5x^2 = 320 \implies x^2 = \frac{320}{5} = 64 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{64} = \pm 8 \][/tex]
So, the solutions are [tex]\(x = 8\)[/tex] and [tex]\(x = -8\)[/tex].
### Quadratic Equation 2: [tex]\(3x^2 = 75\)[/tex]
1. Isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 3x^2 = 75 \implies x^2 = \frac{75}{3} = 25 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{25} = \pm 5 \][/tex]
So, the solutions are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
### Quadratic Equation 3: [tex]\(8x^2 + 9 = 313\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 8x^2 + 9 = 313 \implies 8x^2 = 304 \implies x^2 = \frac{304}{8} = 38 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{38} \approx \pm 6.2 \,(\text{rounded to the nearest tenth}) \][/tex]
So, the solutions are [tex]\(x \approx 6.2\)[/tex] and [tex]\(x \approx -6.2\)[/tex].
### Quadratic Equation 4: [tex]\(32 = 25x^2 - 4\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 32 = 25x^2 - 4 \implies 36 = 25x^2 \implies x^2 = \frac{36}{25} = 1.44 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{1.44} = \pm 1.2 \][/tex]
So, the solutions are [tex]\(x = 1.2\)[/tex] and [tex]\(x = -1.2\)[/tex].
### Quadratic Equation 5: [tex]\(2x^2 - 3 = 29\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 2x^2 - 3 = 29 \implies 2x^2 = 32 \implies x^2 = \frac{32}{2} = 16 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{16} = \pm 4 \][/tex]
So, the solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
### Matching the results:
1. [tex]\(-1 - 5x^2 = -321\)[/tex] - Solutions are [tex]\(x = \pm 8\)[/tex]: Matches with (c) [tex]\(x = 8, -8\)[/tex].
2. [tex]\(3x^2 = 75\)[/tex] - Solutions are [tex]\(x = \pm 5\)[/tex]: Matches with (b) [tex]\(x = 5, -5\)[/tex].
3. [tex]\(8x^2 + 9 = 313\)[/tex] - Solutions are [tex]\(x \approx \pm 6.2\)[/tex]: Matches with (d) [tex]\(x = 6.2, -6.2\)[/tex].
4. [tex]\(32 = 25x^2 - 4\)[/tex] - Solutions are [tex]\(x = \pm 1.2\)[/tex]: Matches with (a) [tex]\(x = 1.2, -1.2\)[/tex].
5. [tex]\(2x^2 - 3 = 29\)[/tex] - Solutions are [tex]\(x = \pm 4\)[/tex]: Matches with (e) [tex]\(x = 4, -4\)[/tex].
Thus, the correct matching is:
1. [tex]\(-1 - 5x^2 = -321\)[/tex] -> (c) [tex]\(x = 8, -8\)[/tex]
2. [tex]\(3x^2 = 75\)[/tex] -> (b) [tex]\(x = 5, -5\)[/tex]
3. [tex]\(8x^2 + 9 = 313\)[/tex] -> (d) [tex]\(x = 6.2, -6.2\)[/tex]
4. [tex]\(32 = 25x^2 - 4\)[/tex] -> (a) [tex]\(x = 1.2, -1.2\)[/tex]
5. [tex]\(2x^2 - 3 = 29\)[/tex] -> (e) [tex]\(x = 4, -4\)[/tex]
### Quadratic Equation 1: [tex]\(-1 - 5x^2 = -321\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ -1 - 5x^2 = -321 \implies -5x^2 = -320 \implies 5x^2 = 320 \implies x^2 = \frac{320}{5} = 64 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{64} = \pm 8 \][/tex]
So, the solutions are [tex]\(x = 8\)[/tex] and [tex]\(x = -8\)[/tex].
### Quadratic Equation 2: [tex]\(3x^2 = 75\)[/tex]
1. Isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 3x^2 = 75 \implies x^2 = \frac{75}{3} = 25 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{25} = \pm 5 \][/tex]
So, the solutions are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
### Quadratic Equation 3: [tex]\(8x^2 + 9 = 313\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 8x^2 + 9 = 313 \implies 8x^2 = 304 \implies x^2 = \frac{304}{8} = 38 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{38} \approx \pm 6.2 \,(\text{rounded to the nearest tenth}) \][/tex]
So, the solutions are [tex]\(x \approx 6.2\)[/tex] and [tex]\(x \approx -6.2\)[/tex].
### Quadratic Equation 4: [tex]\(32 = 25x^2 - 4\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 32 = 25x^2 - 4 \implies 36 = 25x^2 \implies x^2 = \frac{36}{25} = 1.44 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{1.44} = \pm 1.2 \][/tex]
So, the solutions are [tex]\(x = 1.2\)[/tex] and [tex]\(x = -1.2\)[/tex].
### Quadratic Equation 5: [tex]\(2x^2 - 3 = 29\)[/tex]
1. First, isolate the [tex]\(x^2\)[/tex] term:
[tex]\[ 2x^2 - 3 = 29 \implies 2x^2 = 32 \implies x^2 = \frac{32}{2} = 16 \][/tex]
2. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{16} = \pm 4 \][/tex]
So, the solutions are [tex]\(x = 4\)[/tex] and [tex]\(x = -4\)[/tex].
### Matching the results:
1. [tex]\(-1 - 5x^2 = -321\)[/tex] - Solutions are [tex]\(x = \pm 8\)[/tex]: Matches with (c) [tex]\(x = 8, -8\)[/tex].
2. [tex]\(3x^2 = 75\)[/tex] - Solutions are [tex]\(x = \pm 5\)[/tex]: Matches with (b) [tex]\(x = 5, -5\)[/tex].
3. [tex]\(8x^2 + 9 = 313\)[/tex] - Solutions are [tex]\(x \approx \pm 6.2\)[/tex]: Matches with (d) [tex]\(x = 6.2, -6.2\)[/tex].
4. [tex]\(32 = 25x^2 - 4\)[/tex] - Solutions are [tex]\(x = \pm 1.2\)[/tex]: Matches with (a) [tex]\(x = 1.2, -1.2\)[/tex].
5. [tex]\(2x^2 - 3 = 29\)[/tex] - Solutions are [tex]\(x = \pm 4\)[/tex]: Matches with (e) [tex]\(x = 4, -4\)[/tex].
Thus, the correct matching is:
1. [tex]\(-1 - 5x^2 = -321\)[/tex] -> (c) [tex]\(x = 8, -8\)[/tex]
2. [tex]\(3x^2 = 75\)[/tex] -> (b) [tex]\(x = 5, -5\)[/tex]
3. [tex]\(8x^2 + 9 = 313\)[/tex] -> (d) [tex]\(x = 6.2, -6.2\)[/tex]
4. [tex]\(32 = 25x^2 - 4\)[/tex] -> (a) [tex]\(x = 1.2, -1.2\)[/tex]
5. [tex]\(2x^2 - 3 = 29\)[/tex] -> (e) [tex]\(x = 4, -4\)[/tex]
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