Find the real solution(s) of the given equation. $$ (-7 x-1)^{2}-(-7 x-1)-2=0 $$ Give exact answers using fractions and square roots, not de

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
(-7x - 1)^2 - (-7x - 1) - 2 = 0
$$

**Step 1: Substitute for Simplicity**

Let $$ u = -7x - 1 $$. Substituting into the equation gives:

$$
u^2 - u - 2 = 0
$$

**Step 2: Solve the Quadratic Equation**

This is a standard quadratic equation in the form $$ au^2 + bu + c = 0 $$. We can solve for $$ u $$ using the quadratic formula:

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

For our equation, $$ a = 1 $$, $$ b = -1 $$, and $$ c = -2 $$:

$$
u = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)} = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2}
$$

This gives two solutions for $$ u $$:

$$
u = 2 \quad \text{or} \quad u = -1
$$

**Step 3: Back-Substitute to Find $$ x $$**

Now, recall that $$ u = -7x - 1 $$.

1. **Case 1:** $$ u = 2 $$
   $$
   -7x - 1 = 2 \implies -7x = 3 \implies x = -\frac{3}{7}
   $$

2. **Case 2:** $$ u = -1 $$
   $$
   -7x - 1 = -1 \implies -7x = 0 \implies x = 0
   $$

**Conclusion:**

The real solutions to the equation are:

$$
x = 0 \quad \text{and} \quad x = -\frac{3}{7}
$$

**Final Answer:**

After simplifying, the real solutions are x = 0 and x = –3⁄7
The real solutions are $$ x = 0 $$ and $$ x = -\frac{3}{7} $$.

Find the real solution(s) of the given equation. \( (-7 x-1)^{2}-(-7 x-1)-2=0 \) Give exact answers using fractions and square roots, not de