The equation $$ x^{2}+5 x-17=0 $$ has two solutions $$ A $$ and $$ B $$ where $$ A

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UpStudy AI · Accepted Answer

To solve the quadratic equation $$ x^{2} + 5x - 17 = 0 $$, we can use the quadratic formula:

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

where $$ a = 1 $$, $$ b = 5 $$, and $$ c = -17 $$.

### Step-by-Step Solution:

1. **Calculate the discriminant ($$ \Delta $$):**
   
   $$
   \Delta = b^2 - 4ac = 5^2 - 4(1)(-17) = 25 + 68 = 93
   $$

2. **Find the two solutions:**
   
   $$
   x = \frac{-5 \pm \sqrt{93}}{2}
   $$

- **Solution A (smaller root):**
     
     $$
     A = \frac{-5 - \sqrt{93}}{2}
     $$
     
     Calculating the numerical value:
     
     $$
     \sqrt{93} \approx 9.644 \
     A \approx \frac{-5 - 9.644}{2} = \frac{-14.644}{2} \approx -7.322
     $$
   
   - **Solution B (larger root):**
     
     $$
     B = \frac{-5 + \sqrt{93}}{2}
     $$
     
     Calculating the numerical value:
     
     $$
     B \approx \frac{-5 + 9.644}{2} = \frac{4.644}{2} \approx 2.322
     $$

### Final Answers:

- **Exact Expressions:**
  $$
  A = \frac{-5 - \sqrt{93}}{2}, \quad B = \frac{-5 + \sqrt{93}}{2}
  $$

- **Numerical Approximations (to 3 decimal places):**
  $$
  A \approx -7.322, \quad B \approx 2.322
  $$
$$ A \approx -7.322 $$ and $$ B \approx 2.322 $$.

The equation has two solutions and
where
and and
Give your answers to 3 decimal places, or as exact expressions.

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The equation has two solutions and where and and Give your answers to 3 decimal places, or as exact expressions.