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Problem 424

Find the roots of the equation $$x^{2}+6 x+8=0$$

Expert verified
The roots of the equation $$x^{2}+6x+8=0$$ are -2 and -4.
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Step 1: Identify the equation and its coefficients

The given equation is $$x^2 + 6x + 8 = 0$$. The coefficients are: a = 1, b = 6, and c = 8.

Step 2: Factor the quadratic equation

To factor the quadratic equation, we need to find two numbers that multiply to give ac (product) and add up to give b (sum). Product: ac = (1)(8) = 8 Sum: b = 6 The two numbers that meet these conditions are 2 and 4 since (2)(4) = 8 and (2 + 4) = 6. Now, we can rewrite the equation as: $$x^2 + 2x + 4x + 8 = 0$$.

Step 3: Factor by grouping

In this step, we will group the terms and factor out the common factors. Group the terms: $$(x^2 + 2x) + (4x + 8) = 0$$ Factor out the common factors: $$x(x + 2) + 4(x + 2) = 0$$ Now, we see that both terms have a common factor of (x + 2). So, factor (x + 2) from both terms: $$(x + 2)(x + 4) = 0$$

Step 4: Find the roots

To find the roots of the equation, we need to solve for x in each factor. For the first factor: $$x + 2 = 0$$ x = -2 For the second factor: $$x + 4 = 0$$ x = -4 So, the roots of the equation $$x^2 + 6x + 8 = 0$$ are -2 and -4.

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