solve the following quadratic equation 1. 6s2 − 6s − 4 = 0 2. 2x2 + 7x + 5 = 0 3. 4u2 − u − 4 = 0 4. 2z2 + 4z + 2 = 0 5. 4n2 + 9n + 4 = 0 6. 8h2 − 2h − 1 = 0 7. f2 + 7f + 4 = 0 8. 4m2 + 8m + 4 = 0

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01-04-2022
Math

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solve the following quadratic equation 1. 6s2 − 6s − 4 = 0 2. 2x2 + 7x + 5 = 0 3. 4u2 − u − 4 = 0 4. 2z2 + 4z + 2 = 0 5. 4n2 + 9n + 4 = 0 6. 8h2 − 2h − 1 = 0 7. f2 + 7f + 4 = 0 8. 4m2 + 8m + 4 = 0

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NoahBadillaidn NoahBadillaidn · Answer 1 · 2022-04-01T11:16:42+08:00

✒️Quadratic Equation

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Solve the following quadratic equations.

[tex]\rm 1.)\ 6s^{2} - 6s - 4 = 0[/tex]

[tex]\rm 2.)\ 2x^{2} + 7x + 5 = 0[/tex]

[tex]\rm 3.)\ 4u^{2} - u - 4 = 0[/tex]

[tex]\rm 4.)\ 2z^{2} + 4z + 2 = 0[/tex]

[tex]\rm 5.)\ 4n^{2} + 9n + 4 = 0[/tex]

[tex]\rm 6.)\ 8h^{2} - 2h - 1 = 0[/tex]

[tex]\rm 7.)\ f^{2} + 7f + 4 = 0[/tex]

[tex]\rm 8.)\ 4m^{2} + 8m + 4 = 0[/tex]

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[tex]\rm We\ know\ that\ the\ quadratic\ formula\ is:[/tex]

[tex]\boxed{\blue{ \rm \: x=\frac{-b \pm \sqrt{b^{2} - 4ac} }{2a} }}[/tex]

_______________________

[tex]\rm Solve:[/tex]

[tex]\rm 1.)\ 6s^{2} - 6s - 4 = 0[/tex]

[tex]\rm s = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm s = \dfrac{-(-6) \pm \sqrt{(-6)^{2} - 4(6)(-4)} }{2(6)}[/tex]

[tex]\rm s = \dfrac{6 \pm \sqrt{36 + 96} }{12}[/tex]

[tex]\rm s = \dfrac{6 \pm \sqrt{132} }{12}[/tex]

[tex]\rm s = \dfrac{6 + \sqrt{132} }{12}\ \ or\ \ \rm s = \dfrac{6 - \sqrt{132} }{12}[/tex]

[tex]\rm s \approx 1.46\ or\ s \approx 0.46[/tex]

[tex]\rm 2.)\ 2x^{2} + 7x + 5 = 0[/tex]

[tex]\rm x = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm x = \dfrac{-7 \pm \sqrt{7^{2} - 4(2)(5)} }{2(2)}[/tex]

[tex]\rm x = \dfrac{-7 \pm \sqrt{49-40} }{4}[/tex]

[tex]\rm x = \dfrac{-7 \pm \sqrt{9} }{4}[/tex]

[tex]\rm x = \dfrac{-7 + \sqrt{9} }{4}\ or\ \rm x = \dfrac{-7 - \sqrt{9} }{4}[/tex]

[tex]\rm x = \dfrac{-7 + 3}{4}\ or\ \rm x = \dfrac{7 - 3}{4}[/tex]

[tex]\rm x = -1\ or\ x = \dfrac{-5}{2}[/tex]

[tex]\rm 3.)\ 4u^{2} - u - 4 = 0[/tex]

[tex]\rm u = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm u = \dfrac{-(-1) \pm \sqrt{(-1)^{2} -4(4)(-4)} }{2(4)}[/tex]

[tex]\rm u = \dfrac{1 \pm \sqrt{1 + 64} }{8}[/tex]

[tex]\rm u = \dfrac{1 \pm \sqrt{65} }{8}[/tex]

[tex]\rm u = \dfrac{1 + \sqrt{65} }{8}\ or\ \rm u = \dfrac{1 - \sqrt{65} }{8}[/tex]

[tex]\rm u \approx 1.13\ or\ u \approx -0.88[/tex]

[tex]\rm 4.)\ 2z^{2} + 4z + 2 = 0[/tex]

[tex]\rm z = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm z = \dfrac{-4 \pm \sqrt{4^{2} - 4(2)(2)} }{2(2)}[/tex]

[tex]\rm z = \dfrac{-4 \pm \sqrt{16-16} }{4}[/tex]

[tex]\rm z = \dfrac{-4 \pm \sqrt{0} }{4}[/tex]

[tex]\rm z = \dfrac{-4 + \sqrt{0} }{4}\ or\ \rm z = \dfrac{-4 - \sqrt{0} }{4}[/tex]

[tex]\rm z = \dfrac{-4 + 0 }{4}\ or\ \rm z = \dfrac{-4 - 0 }{4}[/tex]

[tex]\rm z = -1\ or\ z = -1[/tex]

[tex]\rm Since\ the\ two\ solutions\ are\ same, they\ should\ be\ written\ as:[/tex]

[tex]\rm z = -1[/tex]

[tex]\rm 5.)\ 4n^{2} + 9n + 4 = 0[/tex]

[tex]\rm n = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm n = \dfrac{-9 \pm \sqrt{9^{2} - 4(4)(4)} }{2(4)}[/tex]

[tex]\rm n = \dfrac{-9 \pm \sqrt{81-64} }{8}[/tex]

[tex]\rm n = \dfrac{-9 \pm \sqrt{17} }{8}[/tex]

[tex]\rm n = \dfrac{-9 + \sqrt{17} }{8}\ or\ \rm n = \dfrac{-9 - \sqrt{17} }{8}[/tex]

[tex]\rm n \approx -0.61\ or\ n \approx -1.64[/tex]

[tex]\rm 6.)\ 8h^{2} - 2h - 1 = 0[/tex]

[tex]\rm h = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm h = \dfrac{-(-2) \pm \sqrt{(-2)^{2} - 4(8)(-1)} }{2(8)}[/tex]

[tex]\rm h = \dfrac{2 \pm \sqrt{4 + 32} }{16}[/tex]

[tex]\rm h = \dfrac{2 \pm \sqrt{36} }{16}[/tex]

[tex]\rm h = \dfrac{2 + \sqrt{36} }{16}\ or\ \rm h = \dfrac{2 - \sqrt{36} }{16}[/tex]

[tex]\rm h = \dfrac{2 + 6}{16}\ or\ \rm h = \dfrac{2 - 6}{16}[/tex]

[tex]\rm h = \dfrac{1}{2}\ or\ h = \dfrac{-1}{4}[/tex]

[tex]\rm 7.)\ f^{2} + 7f + 4 = 0[/tex]

[tex]\rm f = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm f = \dfrac{-7 \pm \sqrt{7^{2} - 4(1)(4)} }{2(1)}[/tex]

[tex]\rm f = \dfrac{-7 \pm \sqrt{49-16} }{2}[/tex]

[tex]\rm f = \dfrac{-7 \pm \sqrt{33} }{2}[/tex]

[tex]\rm f = \dfrac{-7 + \sqrt{33} }{2}\ or\ \rm f = \dfrac{-7 - \sqrt{33} }{2}[/tex]

[tex]\rm f \approx -0.63\ or\ f \approx -6.37[/tex]

[tex]\rm 8.)\ 4m^{2} + 8m + 4 = 0[/tex]

[tex]\rm m = \dfrac{-b \pm \sqrt{b^{2} - 4ac} }{2a}[/tex]

[tex]\rm m = \dfrac{-8 \pm \sqrt{8^{2} - 4(4)(4)} }{2(4)}[/tex]

[tex]\rm m = \dfrac{-8 \pm \sqrt{64-64} }{8}[/tex]

[tex]\rm m = \dfrac{-8 \pm \sqrt{0} }{8}[/tex]

[tex]\rm m = \dfrac{-8 + \sqrt{0} }{8}\ or\ \rm m = \dfrac{-8 - \sqrt{0} }{8}[/tex]

[tex]\rm m = \dfrac{-8 + 0}{8}\ or\ \rm m = \dfrac{-8 - 0}{8}[/tex]

[tex]\rm m = -1\ or\ m = -1[/tex]

[tex]\rm Since\ the\ two\ solutions\ are\ same, they\ should\ be\ written\ as:[/tex]

[tex]\rm m = -1[/tex]

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Sagot :

✒️Quadratic Equation

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