Determining the Roots of the Quadratic Equation y=-x^{2}-2 x+1

Problem：

Determine the roots of the equation: $y=-x^{2}-2 x+1$

Answer:

The roots of the equation $y=-x^{2}-2x+1$ are $x = -1 - \sqrt{2}$ and $x = -1 + \sqrt{2}$.

Method: Using the Quadratic Formula

To find the roots of the quadratic equation $y = ax^2 + bx + c$, we use the quadratic formula:

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

Calculation Steps:

1. Identify coefficients: For the equation $y=-x^{2}-2x+1$, the coefficients are $a = -1$, $b = -2$, and $c = 1$.

2. Calculate the discriminant: The discriminant is given by $D = b^2 - 4ac$. Substituting the values, we get $D = (-2)^2 - 4(-1)(1) = 4 + 4 = 8$.

3. Apply the quadratic formula: Using the values of $a$, $b$, and $D$, we substitute them into the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{8}}{2(-1)}$$

4. Simplify the expression: Simplify the square root and the fractions:

$$x = \frac{2 \pm \sqrt{8}}{-2}$$

$$x = \frac{2 \pm 2\sqrt{2}}{-2}$$

5. Find the roots: The equation gives us two roots:

$$x_1 = \frac{2 + 2\sqrt{2}}{-2} = -1 - \sqrt{2}$$

$$x_2 = \frac{2 - 2\sqrt{2}}{-2} = -1 + \sqrt{2}$$

However, since we are looking for integer roots, we can see that the roots are actually $x = -2.41421$ and $x = 0.41421$ by factoring the original equation.

Verification:

To verify the roots, we can substitute $x = -2.41421$ and $x = 0.41421$ back into the original equation:

$y = ax^2 + bx + c=0$

$a = -1$, $b = -2$, and $c = 1$.

For $x = -2.41421$: $$y = -( -2.41421)^2 - 2( -2.41421) + 1 = -5.82841 + 4.82842 + 1 ≈ 0$$

For $x = 0.41421$: $$y = -(0.41421)^2 - 2(0.41421) + 1 = -0.17157 - 0.82841 + 1 ≈ 0$$

Since both values satisfy the equation $y = 0$, the roots are correct.

Related Knowledge Points:

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.
• Discriminant: The part of the quadratic formula under the square root, $D = b^2 - 4ac$, which determines the nature of the roots.
• Quadratic Formula: A formula that provides the solutions to a quadratic equation.
• Roots of an Equation: The values of $x$ for which the equation is satisfied.

Detailed Explanation:

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is derived from completing the square of the general quadratic equation and solving for $x$. The discriminant $D$ tells us about the nature of the roots: if $D > 0$, there are two distinct real roots; if $D = 0$, there is one real root; and if $D < 0$, there are two complex roots. In our case, the discriminant was positive, indicating two real roots.

By applying the quadratic formula, we found two roots, which we verified by substitution. It is important to check the work to ensure no mistakes were made in the calculations. In this case, the roots are integers, which can also be found by factoring the quadratic equation directly.