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.

Upstudy AI Solution

Answer

Solution

a)

b)

c)

d)

e)

f)

Summary of Roots:

Beyond the Answer

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