To solve the equation [tex]\((2 x + \sqrt{2})(5 x + \sqrt{19}) = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that make the equation true.
The given equation can be solved by applying the zero-product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we can set each factor in the equation to zero and solve for [tex]\(x\)[/tex] separately.
Let's proceed step by step:
1. Set each factor equal to zero and solve:
- First factor:
[tex]\[
2 x + \sqrt{2} = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
2 x = -\sqrt{2}
\][/tex]
[tex]\[
x = -\frac{\sqrt{2}}{2}
\][/tex]
- Second factor:
[tex]\[
5 x + \sqrt{19} = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
5 x = -\sqrt{19}
\][/tex]
[tex]\[
x = -\frac{\sqrt{19}}{5}
\][/tex]
2. Write down the solutions:
The solutions to the equation [tex]\((2 x + \sqrt{2})(5 x + \sqrt{19}) = 0\)[/tex] are:
[tex]\[
x = -\frac{\sqrt{2}}{2} \quad \text{and} \quad x = -\frac{\sqrt{19}}{5}
\][/tex]
In conclusion, the solutions to the given equation are [tex]\(x = -\frac{\sqrt{2}}{2}\)[/tex] and [tex]\(x = -\frac{\sqrt{19}}{5}\)[/tex].
