Answer:
Second statement 2x2 + 8x – 24 = 0 Is true for the given conditions. When x = -6 2x2 + 8x – 24 = 0 Becomes 2(-6)2 + 8(-6) – 24 = 0 2(36) - 48 - 24 = 0 72 - 48 - 24 = 0 0 = 0 Which is true. When x = 2 2x2 + 8x – 24 = 0 Becomes 2(2)2 + 8(2) – 24 = 0 2(4) + 16 - 24 = 0 8 + 16 - 24 = 0 0 = 0 Which is true. So 2x2 + 8x – 24 = 0 will be answer.
The characteristics of the solutions of the polynomials are listed below:
Let be a second order polynomial of the form [tex]a\cdot x^{2}+b\cdot x + c = 0[/tex], there are no real roots if and only if [tex]b^{2}-4\cdot a \cdot c < 0[/tex]. Now we proceed to determine the nature of the roots of each polynomial:
1) [tex]x^{2} = 49[/tex]
This expression is equivalent to the polynomial [tex]x^{2}-49 = 0[/tex], then the discriminant is:
[tex]d = 0^{2}-4\cdot (1)\cdot (-49)[/tex]
[tex]d = 196[/tex]
The polynomial has two distinct real roots.
2) [tex]2\cdot x^{2} = -128[/tex]
This expression is equivalent to the polynomial [tex]2\cdot x^{2} +128 = 0[/tex], then the discriminant is:
[tex]d = 0^{2} - 4\cdot (2)\cdot (128)[/tex]
[tex]d = -1024[/tex]
The polynomial has two conjugated complex roots.
3) [tex]3\cdot x^{2}-16 = 32[/tex]
This expression is equivalent to the polynomial [tex]3\cdot x^{2}-48 = 0[/tex], then the discriminant is:
[tex]d = 0^{2}-4\cdot (3) \cdot (-48)[/tex]
[tex]d = 576[/tex]
The polynomial has two distinct real roots.
4) [tex]x^{2}+x+1 = 0[/tex]
The discriminant of this expression is:
[tex]d = 1^{2}-4\cdot (1)\cdot (1)[/tex]
[tex]d = -3[/tex]
The polynomial has two conjugated complex roots.
Nota - The original statement reports typing mistakes. Corrected form is presented below:
Select all the equations that do not have a real solution:
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