Questions: x^2 + 4x + 7 = -1

x^2 + 4x + 7 = -1
Transcript text: $x^{2}+4 x+7=-1$
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Solution

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Solution Steps

To solve the quadratic equation \(x^2 + 4x + 7 = -1\), we first move all terms to one side to set the equation to zero. This gives us \(x^2 + 4x + 8 = 0\). We can then use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots of the equation, where \(a = 1\), \(b = 4\), and \(c = 8\).

Step 1: Set the Equation to Zero

We start with the equation: \[ x^2 + 4x + 7 = -1 \] By moving \(-1\) to the left side, we rewrite the equation as: \[ x^2 + 4x + 8 = 0 \]

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ D = b^2 - 4ac \] Substituting \(a = 1\), \(b = 4\), and \(c = 8\): \[ D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16 \] Since the discriminant is negative, the solutions will be complex.

Step 3: Find the Roots

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting \(D = -16\): \[ x = \frac{-4 \pm \sqrt{-16}}{2 \cdot 1} = \frac{-4 \pm 4i}{2} = -2 \pm 2i \] Thus, the two solutions are: \[ x_1 = -2 + 2i \quad \text{and} \quad x_2 = -2 - 2i \]

Final Answer

The solutions to the equation \(x^2 + 4x + 7 = -1\) are: \[ \boxed{x = -2 + 2i} \quad \text{and} \quad \boxed{x = -2 - 2i} \]

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