Identify a method that could be used to determine the roots of the given equations. Then use it to determine the roots. $$ \begin{array}{ll} ext { a) } x^{2}=15 x & ext { d) }(x-5)(2 x-3)=(x+4)(x-3 \ ext { b) } x^{2}=115 & ext { e) }-2(x-3)^{2}+50=0 \ ext { c) } 2 x^{2}=19 x-24 & ext { f) } 1.5 x^{2}-26.7 x+2.4=0\end{array} $$

Question

Identify a method that could be used to determine the roots of the given equations. Then use it to determine the roots. $$ \begin{array}{ll}	ext { a) } x^{2}=15 x & 	ext { d) }(x-5)(2 x-3)=(x+4)(x-3 \ 	ext { b) } x^{2}=115 & 	ext { e) }-2(x-3)^{2}+50=0 \ 	ext { c) } 2 x^{2}=19 x-24 & 	ext { f) } 1.5 x^{2}-26.7 x+2.4=0\end{array} $$

UpStudy AI · Accepted Answer

To determine the roots of the given equations, we can use various methods such as factoring, the quadratic formula, or completing the square. Below, I will identify a suitable method for each equation and then solve for the roots.

### a) $$ x^{2} = 15x $$

**Method:** Rearranging and factoring.

**Solution:**
1. Rearrange the equation: $$ x^2 - 15x = 0 $$
2. Factor: $$ x(x - 15) = 0 $$
3. Set each factor to zero: 
   - $$ x = 0 $$
   - $$ x - 15 = 0 $$ → $$ x = 15 $$

**Roots:** $$ x = 0, 15 $$

---

### b) $$ x^{2} = 115 $$

**Method:** Taking the square root.

**Solution:**
1. Rearrange the equation: $$ x^2 - 115 = 0 $$
2. Take the square root: $$ x = \pm \sqrt{115} $$

**Roots:** $$ x = \sqrt{115}, -\sqrt{115} $$ (approximately $$ x \approx 10.72, -10.72 $$)

---

### c) $$ 2x^{2} = 19x - 24 $$

**Method:** Rearranging and using the quadratic formula.

**Solution:**
1. Rearrange the equation: $$ 2x^2 - 19x + 24 = 0 $$
2. Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
   - Here, $$ a = 2, b = -19, c = 24 $$
   - Calculate the discriminant: $$ b^2 - 4ac = (-19)^2 - 4(2)(24) = 361 - 192 = 169 $$
   - Calculate the roots:
   $$
   x = \frac{19 \pm \sqrt{169}}{4} = \frac{19 \pm 13}{4}
   $$
   - This gives:
   - $$ x = \frac{32}{4} = 8 $$
   - $$ x = \frac{6}{4} = 1.5 $$

**Roots:** $$ x = 8, 1.5 $$

---

### d) $$ (x-5)(2x-3) = (x+4)(x-3) $$

**Method:** Expanding and rearranging.

**Solution:**
1. Expand both sides:
   - Left: $$ 2x^2 - 3x - 10 $$
   - Right: $$ x^2 + x - 12 $$
2. Set the equation: $$ 2x^2 - 3x - 10 = x^2 + x - 12 $$
3. Rearrange: $$ 2x^2 - x^2 - 3x - x - 10 + 12 = 0 $$
   - This simplifies to: $$ x^2 - 4x + 2 = 0 $$
4. Use the quadratic formula:
   - Here, $$ a = 1, b = -4, c = 2 $$
   - Discriminant: $$ (-4)^2 - 4(1)(2) = 16 - 8 = 8 $$
   - Roots:
   $$
   x = \frac{4 \pm \sqrt{8}}{2} = \frac{4 \pm 2\sqrt{2}}{2} = 2 \pm \sqrt{2}
   $$

**Roots:** $$ x = 2 + \sqrt{2}, 2 - \sqrt{2} $$

---

### e) $$ -2(x-3)^{2} + 50 = 0 $$

**Method:** Rearranging and taking the square root.

**Solution:**
1. Rearrange: $$ -2(x-3)^2 = -50 $$ → $$ (x-3)^2 = 25 $$
2. Take the square root: $$ x - 3 = \pm 5 $$
3. Solve for $$ x $$:
   - $$ x - 3 = 5 $$ → $$ x = 8 $$
   - $$ x - 3 = -5 $$ → $$ x = -2 $$

**Roots:** $$ x = 8, -2 $$

---

### f) $$ 1.5x^{2} - 26.7x + 2.4 = 0 $$

**Method:** Using the quadratic formula.

**Solution:**
1. Use the quadratic formula:
   - Here, $$ a = 1.5, b = -26.7, c = 2.4 $$
   - Discriminant: $$ (-26.7)^2 - 4(1.5)(2.4) = 712.89 - 14.4 = 698.49 $$
   - Roots:
   $$
   x = \frac{26.7 \pm \sqrt{698.49}}{3} \approx \frac{26.7 \pm 26.4}{3}
   $$
   - This gives:
   - $$ x \approx \frac{53.1}{3} \approx 17.7 $$
   - $$ x \approx \frac{0.3}{3} \approx 0.1 $$

**Roots:** $$ x \approx 17.7, 0.1 $$

---

### Summary of Roots:
- a) $$ x = 0, 15 $$
- b) $$ x = \sqrt{115}, -\sqrt{115} $$
- c) $$ x = 8, 1.5 $$
- d) $$ x = 2 + \sqrt{2}, 2 - \sqrt{2} $$
- e) $$ x = 8, -2 $$
- f) $$ x \approx 17.7, 0.1 $$
a) $$ x = 0, 15 $$ b) $$ x = \sqrt{115}, -\sqrt{115} $$ c) $$ x = 8, 1.5 $$ d) $$ x = 2 + \sqrt{2}, 2 - \sqrt{2} $$ e) $$ x = 8, -2 $$ f) $$ x \approx 17.7, 0.1 $$

Identify a method that could be used to determine the roots of the
given equations. Then use it to determine the roots.

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Solución

a)

b)

c)

d)

e)

f)

Summary of Roots:

Beyond the Answer

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