I would really appreciate a brainliest for this, it took me quite a while : )
Answer:
1.) imaginary; x ∉ R
2.) [tex]x=\frac{-17\frac{+}{}\sqrt{399} }{5}[/tex]
3.) [tex]x_1=1;x_2=\frac{1}{4}[/tex]
4.) [tex]x_1=-10+4\sqrt {6}; x_2 =-10-4\sqrt {6}[/tex]
Step-by-step explanation:
1.) 4x² - 2x(3x + 1) = 5
Distribute.
4x² - 6x² + 2x = 5
Combine like terms, change the signs.
-2x² + 2x = 5 ⇒ -2x² + 2x - 5 = 0
Insert into the quadratic formula.
[tex]x=\frac{-b\frac{+}{} \sqrt{b^2-4ac} }{2a}[/tex]; a = -2; b = 2; c = -5
Plug in values for a, b, & c.
[tex]x=\frac{-2\frac{+}{} \sqrt{2^2-4(-2)(-5)} }{2(-2)}[/tex]
Multiply and add/subtract.
[tex]x=\frac{-2\frac{+}{} \sqrt{4-40} }{-4}[/tex] ⇒ [tex]x=\frac{-2\frac{+}{} \sqrt{-36} }{-4}[/tex]
The [tex]\sqrt{-36}[/tex] is imaginary, therefore x ∉ R
2.) -5x(x + 6) = 4(x - 3) - 10
Distribute, combine like terms.
-5x² - 30x = 4x - 22
Change the signs.
5x² + 34x - 22 = 0; a = 5; b = 34; c = -22
Plug in values for a, b, & c in the quadratic formula.
[tex]x=\frac{-34\frac{+}{} \sqrt{34^2-4(5)(-22)} }{2(5)}[/tex]
Multiply and add/subtract, simplify.
[tex]x=\frac{-34\frac{+}{} \sqrt{1156+440} }{10}=\frac{-34\frac{+}{} \sqrt{1596} }{10}x=\frac{-34\frac{+}{} \sqrt{2*2*3*7*19} }{10} =\frac{-34\frac{+}{} 2\sqrt{399} }{10}=\frac{-17\frac{+}{} \sqrt{399} }{5}[/tex]
3.) x² + (1 - x)(1 - 3x) = x
Multiply, combine like terms.
x² + 1 - 3x - x + 3x² = x ⇒ 4x² - 4x + 1 = x
Change the signs.
4x² - 5x + 1 = 0; a = 4; b = -5; c = 1
Insert a, b, & c into quadratic formula.
[tex]x=\frac{-(-5)\frac{+}{} \sqrt{-5^2-4(4)(1)} }{2(4)}[/tex]
Multiply, add, subtract, simplify.
[tex]x=\frac{5\frac{+}{} \sqrt{25-16} }{8} = \frac{5\frac{+}{} \sqrt{9} }{8} = \frac{5\frac{+}{} 3 } {8}[/tex]
[tex]x_1=\frac{5+3}{8} =\frac{8}{8}=1;x_2=\frac{5-3}{8}=\frac{2}{8}=\frac{1}{4}[/tex]
4. (x - 8)(2x + 3) = (3x - 5)(x + 4)
Multiply, combine like terms.
2x² + 3x - 16x - 24 = 3x² + 12x - 5x - 20 ⇒ 2x² - 13x -24 = 3x² + 7x - 20
Change the signs.
x² + 20x + 4 = 0; a = 1; b = 20; c = 4
Insert a, b, & c into quadratic formula.
[tex]x=\frac{-20\frac{+}{} \sqrt{20^2-4(1)(4)} }{2(1)}[/tex]
Multiply, add, subtract, simplify.
[tex]x=\frac{-20\frac{+}{} \sqrt{400-16} }{2}=\frac{-20\frac{+}{} \sqrt{384} }{2}=\frac{-20\frac{+}{} \sqrt{2^6*2*3} }{2}=\frac{-20\frac{+}{} 8\sqrt{6} }{2} = -10\pm4\sqrt {6}[/tex]
[tex]x_1=-10+4\sqrt {6}; x_2 =-10-4\sqrt {6}[/tex]
