Answer:
[tex]a)\ x_1=-3i,\ x_2 =3i[/tex]
[tex]b)\ x_1=-\sqrt{10},\ x_2=\sqrt{10}[/tex]
Step-by-step explanation:
Given equations:
[tex]a)\ x^2+19=10\\\\b)\ 4x^2+3=43[/tex]
A. x² + 19 = 10
Step 1: Subtract 19 from both sides.
[tex]\\\implies x^2+19-19=10-19\\\\\implies x^2=-9[/tex]
Imaginary number rule: For any positive real number "k", [tex]\sqrt{-k} = i\sqrt{k}[/tex]
Note: Two imaginary (complex) solutions indicate that the graph will not intersect the x-axis. As a result, it has no real roots.
Step 2: Take the square root of both sides (positive and negative roots).
[tex]\\\implies \sqrt{x^2}=\sqrt{-9}\\\\\implies x=\pm\ i\sqrt{9}\\\\\implies x=\pm\ 3i[/tex]
Step 3: Separate the solutions.
[tex]\implies x_1=-3i,\ x_2 =3i[/tex]
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B. 4x² + 3 = 43
Step 1: Subtract 43 from both sides.
[tex]\\\implies 4x^2++3-3=43-3\\\\\implies 4x^2=40[/tex]
Step 2: Divide both sides by 4.
[tex]\\\implies 4x^2=40\\\\\implies \dfrac{4x^2}4=\dfrac{40}{4}\\\\\implies x^2=10[/tex]
Step 3: Take the square root of both sides (positive and negative roots).
[tex]\\\implies \sqrt{x^2}=\sqrt{10}\\\\\implies x=\pm\ \sqrt{10}[/tex]
Step 4: Separate the solutions.
[tex]\implies x_1=-\sqrt{10},\ x_2=\sqrt{10}[/tex]
