Pregunta
1. \( \frac{x^{2}}{16}-\frac{1}{25} \) 2. \( 15 x^{2}-36 x+12 \) 3. \( 2 x^{2}-7 x-4 \) 4. \( 3 x^{2}-8 x+4 \) 5. \( 6 x^{2}+4 x-2 \) 6. \( 2 x^{2}+3 x-5 \) 7. \( 9 x^{2}+27 x+8 \) 8. \( 12 x^{2}+4 x-5 \) 9. \( 24 x^{2}+x-3 \)
Ask by Byrd Morgan. in South Africa
Mar 16,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
Here are the factorizations for the given expressions:
1. \( \frac{x^{2}}{16}-\frac{1}{25} \) – No factorization found.
2. \( 15 x^{2}-36 x+12 \) – \( 3(x-2)(5x-2) \)
3. \( 2 x^{2}-7 x-4 \) – \( (x-4)(2x+1) \)
4. \( 3 x^{2}-8 x+4 \) – \( (x-2)(3x-2) \)
5. \( 6 x^{2}+4 x-2 \) – \( 2(x+1)(3x-1) \)
6. \( 2 x^{2}+3 x-5 \) – \( (x-1)(2x+5) \)
7. \( 9 x^{2}+27 x+8 \) – \( (3x+1)(3x+8) \)
8. \( 12 x^{2}+4 x-5 \) – \( (2x-1)(6x+5) \)
9. \( 24 x^{2}+x-3 \) – \( (3x-1)(8x+3) \)
Solución
Factor the expression by following steps:
- step0: Factor:
\(3x^{2}-8x+4\)
- step1: Rewrite the expression:
\(3x^{2}+\left(-2-6\right)x+4\)
- step2: Calculate:
\(3x^{2}-2x-6x+4\)
- step3: Rewrite the expression:
\(x\times 3x-x\times 2-2\times 3x+2\times 2\)
- step4: Factor the expression:
\(x\left(3x-2\right)-2\left(3x-2\right)\)
- step5: Factor the expression:
\(\left(x-2\right)\left(3x-2\right)\)
Factor the expression \( 24 x^{2}+x-3 \).
Factor the expression by following steps:
- step0: Factor:
\(24x^{2}+x-3\)
- step1: Rewrite the expression:
\(24x^{2}+\left(9-8\right)x-3\)
- step2: Calculate:
\(24x^{2}+9x-8x-3\)
- step3: Rewrite the expression:
\(3x\times 8x+3x\times 3-8x-3\)
- step4: Factor the expression:
\(3x\left(8x+3\right)-\left(8x+3\right)\)
- step5: Factor the expression:
\(\left(3x-1\right)\left(8x+3\right)\)
Factor the expression \( 2 x^{2}+3 x-5 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{2}+3x-5\)
- step1: Rewrite the expression:
\(2x^{2}+\left(5-2\right)x-5\)
- step2: Calculate:
\(2x^{2}+5x-2x-5\)
- step3: Rewrite the expression:
\(x\times 2x+x\times 5-2x-5\)
- step4: Factor the expression:
\(x\left(2x+5\right)-\left(2x+5\right)\)
- step5: Factor the expression:
\(\left(x-1\right)\left(2x+5\right)\)
Factor the expression \( 6 x^{2}+4 x-2 \).
Factor the expression by following steps:
- step0: Factor:
\(6x^{2}+4x-2\)
- step1: Rewrite the expression:
\(2\times 3x^{2}+2\times 2x-2\)
- step2: Factor the expression:
\(2\left(3x^{2}+2x-1\right)\)
- step3: Factor the expression:
\(2\left(x+1\right)\left(3x-1\right)\)
Factor the expression \( 15 x^{2}-36 x+12 \).
Factor the expression by following steps:
- step0: Factor:
\(15x^{2}-36x+12\)
- step1: Rewrite the expression:
\(3\times 5x^{2}-3\times 12x+3\times 4\)
- step2: Factor the expression:
\(3\left(5x^{2}-12x+4\right)\)
- step3: Factor the expression:
\(3\left(x-2\right)\left(5x-2\right)\)
Factor the expression \( 12 x^{2}+4 x-5 \).
Factor the expression by following steps:
- step0: Factor:
\(12x^{2}+4x-5\)
- step1: Rewrite the expression:
\(12x^{2}+\left(10-6\right)x-5\)
- step2: Calculate:
\(12x^{2}+10x-6x-5\)
- step3: Rewrite the expression:
\(2x\times 6x+2x\times 5-6x-5\)
- step4: Factor the expression:
\(2x\left(6x+5\right)-\left(6x+5\right)\)
- step5: Factor the expression:
\(\left(2x-1\right)\left(6x+5\right)\)
Factor the expression \( 9 x^{2}+27 x+8 \).
Factor the expression by following steps:
- step0: Factor:
\(9x^{2}+27x+8\)
- step1: Rewrite the expression:
\(9x^{2}+\left(24+3\right)x+8\)
- step2: Calculate:
\(9x^{2}+24x+3x+8\)
- step3: Rewrite the expression:
\(3x\times 3x+3x\times 8+3x+8\)
- step4: Factor the expression:
\(3x\left(3x+8\right)+3x+8\)
- step5: Factor the expression:
\(\left(3x+1\right)\left(3x+8\right)\)
Factor the expression \( 2 x^{2}-7 x-4 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{2}-7x-4\)
- step1: Rewrite the expression:
\(2x^{2}+\left(1-8\right)x-4\)
- step2: Calculate:
\(2x^{2}+x-8x-4\)
- step3: Rewrite the expression:
\(x\times 2x+x-4\times 2x-4\)
- step4: Factor the expression:
\(x\left(2x+1\right)-4\left(2x+1\right)\)
- step5: Factor the expression:
\(\left(x-4\right)\left(2x+1\right)\)
Here are the factorizations for the given expressions:
1. **Expression:** \( \frac{x^{2}}{16}-\frac{1}{25} \)
**Factorization:** No result could be found for this input.
2. **Expression:** \( 15 x^{2}-36 x+12 \)
**Factorization:** \( 3(x-2)(5x-2) \)
3. **Expression:** \( 2 x^{2}-7 x-4 \)
**Factorization:** \( (x-4)(2x+1) \)
4. **Expression:** \( 3 x^{2}-8 x+4 \)
**Factorization:** \( (x-2)(3x-2) \)
5. **Expression:** \( 6 x^{2}+4 x-2 \)
**Factorization:** \( 2(x+1)(3x-1) \)
6. **Expression:** \( 2 x^{2}+3 x-5 \)
**Factorization:** \( (x-1)(2x+5) \)
7. **Expression:** \( 9 x^{2}+27 x+8 \)
**Factorization:** \( (3x+1)(3x+8) \)
8. **Expression:** \( 12 x^{2}+4 x-5 \)
**Factorization:** \( (2x-1)(6x+5) \)
9. **Expression:** \( 24 x^{2}+x-3 \)
**Factorization:** \( (3x-1)(8x+3) \)
If you need further assistance or explanations on any of these factorizations, feel free to ask!
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución Extra Insights
For the first expression, \( \frac{x^{2}}{16}-\frac{1}{25} \), you may notice that it's structured like the difference of two squares. This means it can be factored or rearranged to find roots more easily. By rewriting it in a more recognizable format, you can derive values of \( x \) where this expression equals zero, which can help in real-world scenarios, like optimizing area in geometry. Moving on to quadratic expressions such as \( 15x^{2}-36x+12 \), they often represent parabolas in coordinate geometry. In practical terms, these equations can model various phenomena, from projectile motion to profit maximization in economics. Being able to factor these or apply the quadratic formula lets you uncover important insights, such as the vertex (which provides maximal or minimal value) or roots (when the graph intersects the x-axis).
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