$$ x = \frac { - 6 \pm \sqrt { 6 ^ { 2 } - 4 ( 1 ) } } { 2 ( 1 ) } $$

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UpStudy AI · Accepted Answer

Certainly! Let's simplify the given quadratic formula step by step.

### Given Expression:
$$
x = \frac{ -6 \pm \sqrt{6^2 - 4 \cdot 1 }}{2 \cdot 1}
$$

### Step 1: Simplify Inside the Square Root
Calculate the discriminant ($$ \Delta $$):
$$
\Delta = b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot 1 = 36 - 4 = 32
$$

### Step 2: Substitute Back into the Formula
$$
x = \frac{ -6 \pm \sqrt{32} }{2}
$$

### Step 3: Simplify the Square Root
$$
\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}
$$

### Step 4: Substitute and Simplify the Expression
$$
x = \frac{ -6 \pm 4\sqrt{2} }{2} = \frac{ -6 }{2} \pm \frac{4\sqrt{2}}{2} = -3 \pm 2\sqrt{2}
$$

### Final Solutions:
$$
x = -3 + 2\sqrt{2} \quad \text{and} \quad x = -3 - 2\sqrt{2}
$$

### Approximate Numerical Values:
If you'd like approximate decimal values:

$$
\sqrt{2} \approx 1.4142
$$
$$
x \approx -3 + 2(1.4142) = -3 + 2.8284 = -0.1716
$$
$$
x \approx -3 - 2.8284 = -5.8284
$$

### Conclusion:
The solutions to the quadratic equation are:
$$
x = -3 + 2\sqrt{2} \quad \text{and} \quad x = -3 - 2\sqrt{2}
$$
Or, approximately:
$$
x \approx -0.1716 \quad \text{and} \quad x \approx -5.8284
$$
The solutions are $$ x = -3 + 2\sqrt{2} $$ and $$ x = -3 - 2\sqrt{2} $$.

\( x = \frac { - 6 \pm \sqrt { 6 ^ { 2 } - 4 ( 1 ) } } { 2 ( 1 ) } \)

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