Question

Question

Watch Video

Show Examples
Solve the equation x, squared, plus, 2, x, minus, 17, equals, 0x
2
+2x−17=0 to the nearest tenth.
Answer
Attempt 2 out of 5

Additional Solution
No Solution
x, equalsx=

Answer 1

To solve the quadratic equation \( x^2 + 2x - 17 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 1 \), \( b = 2 \), and \( c = -17 \).

First, we calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-17) = 4 + 68 = 72
\]

Next, we substitute the values into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{72}}{2 \cdot 1}
\]

Simplifying \( \sqrt{72} \):

\[
\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}
\]

Now, substitute back into the formula:

\[
x = \frac{-2 \pm 6\sqrt{2}}{2}
\]

We can split this into two solutions:

\[
x = \frac{-2 + 6\sqrt{2}}{2} \quad \text{and} \quad x = \frac{-2 - 6\sqrt{2}}{2}
\]

Simplifying each of these:

1. For \( x = \frac{-2 + 6\sqrt{2}}{2} \):

\[
x = -1 + 3\sqrt{2}
\]

2. For \( x = \frac{-2 - 6\sqrt{2}}{2} \):

\[
x = -1 - 3\sqrt{2}
\]

Now, we need to calculate these values. First, we'll find \( 3\sqrt{2} \) and then each solution:

\[
\sqrt{2} \approx 1.414
\]

\[
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242
\]

Now, substituting \( 3\sqrt{2} \) into our solutions:

1. \( x = -1 + 3\sqrt{2} \):

\[
x \approx -1 + 4.242 \approx 3.242 \quad \Rightarrow \quad \text{rounded to nearest tenth: } 3.2
\]

2. \( x = -1 - 3\sqrt{2} \):

\[
x \approx -1 - 4.242 \approx -5.242 \quad \Rightarrow \quad \text{rounded to nearest tenth: } -5.2
\]

Thus, the solution to the equation \( x^2 + 2x - 17 = 0 \) to the nearest tenth is:

**x = 3.2** and **x = -5.2**.