Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real
solutions, enter “DNE” in both boxes.

Question

Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real solutions, enter "DNE" in both boxes. $$ 1+x-x^{2}=0 $$

UpStudy AI · Accepted Answer

To solve the quadratic equation $$ 1 + x - x^{2} = 0 $$ using the quadratic formula, follow these steps:

1. **Rewrite the Equation in Standard Form:**
   $$
   -x^{2} + x + 1 = 0 \quad \text{or} \quad x^{2} - x - 1 = 0
   $$
   Here, $$ a = 1 $$, $$ b = -1 $$, and $$ c = -1 $$.

2. **Apply the Quadratic Formula:**
   $$
   x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
   $$
   Substitute the values of $$ a $$, $$ b $$, and $$ c $$:
   $$
   x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}
   $$

3. **Determine the Real Solutions:**
   $$
   x = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{5}}{2}
   $$
   Both solutions are real numbers since the discriminant ($$ \sqrt{5} $$) is positive.

**Final Answer:**
$$
x = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{5}}{2}
$$
The solutions are $$ x = \frac{1 + \sqrt{5}}{2} $$ and $$ x = \frac{1 - \sqrt{5}}{2} $$.

Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real
solutions, enter “DNE” in both boxes.

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Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real solutions, enter “DNE” in both boxes.

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Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real
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