$$ \begin{array}{l} 2 x^{2}+3 x-1=0 \ 2 x^{2}-30 x+25=0 \end{array} {} x^{2}-23 x+120=0 {} 16 x^{2}-24 x+9=0 $$

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UpStudy AI · Accepted Answer

Sure! Let's solve each of the quadratic equations you've provided. We'll use the quadratic formula for each, which is:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

where $$ a $$, $$ b $$, and $$ c $$ are coefficients from the quadratic equation $$ ax^2 + bx + c = 0 $$.

---

### 1. $$ 2x^{2} + 3x - 1 = 0 $$

**Coefficients:**
- $$ a = 2 $$
- $$ b = 3 $$
- $$ c = -1 $$

**Discriminant ($$ D $$):**
$$
D = b^2 - 4ac = 3^2 - 4(2)(-1) = 9 + 8 = 17
$$

**Solutions:**
$$
x = \frac{-3 \pm \sqrt{17}}{2 \times 2} = \frac{-3 \pm \sqrt{17}}{4}
$$

**Approximate Values:**
$$
x \approx \frac{-3 + 4.1231}{4} \approx \frac{1.1231}{4} \approx 0.2808
$$
$$
x \approx \frac{-3 - 4.1231}{4} \approx \frac{-7.1231}{4} \approx -1.7808
$$

---

### 2. $$ 2x^{2} - 30x + 25 = 0 $$

**Coefficients:**
- $$ a = 2 $$
- $$ b = -30 $$
- $$ c = 25 $$

**Discriminant ($$ D $$):**
$$
D = (-30)^2 - 4(2)(25) = 900 - 200 = 700
$$

**Solutions:**
$$
x = \frac{30 \pm \sqrt{700}}{2 \times 2} = \frac{30 \pm 10\sqrt{7}}{4} = \frac{15 \pm 5\sqrt{7}}{2}
$$

**Approximate Values:**
$$
\sqrt{7} \approx 2.6458
$$
$$
x \approx \frac{30 + 26.4583}{4} = \frac{56.4583}{4} \approx 14.1146
$$
$$
x \approx \frac{30 - 26.4583}{4} = \frac{3.5417}{4} \approx 0.8854
$$

---

### 3. $$ x^{2} - 23x + 120 = 0 $$

**Coefficients:**
- $$ a = 1 $$
- $$ b = -23 $$
- $$ c = 120 $$

**Discriminant ($$ D $$):**
$$
D = (-23)^2 - 4(1)(120) = 529 - 480 = 49
$$

**Solutions:**
$$
x = \frac{23 \pm \sqrt{49}}{2 \times 1} = \frac{23 \pm 7}{2}
$$

**Values:**
$$
x = \frac{23 + 7}{2} = \frac{30}{2} = 15
$$
$$
x = \frac{23 - 7}{2} = \frac{16}{2} = 8
$$

---

### 4. $$ 16x^{2} - 24x + 9 = 0 $$

**Coefficients:**
- $$ a = 16 $$
- $$ b = -24 $$
- $$ c = 9 $$

**Discriminant ($$ D $$):**
$$
D = (-24)^2 - 4(16)(9) = 576 - 576 = 0
$$

Since the discriminant is zero, there is **one real repeated root**.

**Solution:**
$$
x = \frac{24}{2 \times 16} = \frac{24}{32} = \frac{3}{4}
$$

**Alternatively, Factoring:**
$$
16x^2 - 24x + 9 = (4x - 3)^2 = 0 \implies 4x - 3 = 0 \implies x = \frac{3}{4}
$$

---

### Summary of Solutions

1. **$$ 2x^{2} + 3x - 1 = 0 $$**
   - $$ x = \frac{-3 \pm \sqrt{17}}{4} $$
   - Approximately $$ x \approx 0.2808 $$ and $$ x \approx -1.7808 $$

2. **$$ 2x^{2} - 30x + 25 = 0 $$**
   - $$ x = \frac{15 \pm 5\sqrt{7}}{2} $$
   - Approximately $$ x \approx 14.1146 $$ and $$ x \approx 0.8854 $$

3. **$$ x^{2} - 23x + 120 = 0 $$**
   - $$ x = 15 $$ and $$ x = 8 $$

4. **$$ 16x^{2} - 24x + 9 = 0 $$**
   - $$ x = \frac{3}{4} $$

If you have any further questions or need additional explanations for any of these solutions, feel free to ask!
The solutions to the equations are: 1. $$ x = \frac{-3 \pm \sqrt{17}}{4} $$ (approximately $$ x \approx 0.2808 $$ and $$ x \approx -1.7808 $$) 2. $$ x = \frac{15 \pm 5\sqrt{7}}{2} $$ (approximately $$ x \approx 14.1146 $$ and $$ x \approx 0.8854 $$) 3. $$ x = 15 $$ and $$ x = 8 $$ 4. $$ x = \frac{3}{4} $$

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