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Sagot :
[tex] \sf \: {x}^{2} + 3x + 5 = 0[/tex]
Solve the quadratic equation (ax²+bx+c=0) using the quadratic formula:
[tex] \boxed{ \bf \: x = \frac{ - b± \sqrt{ {b}^{2} - 4ac } }{2a} }[/tex]
- a = 1
- b = 3
- c = 5
[tex] \tt \: x = \frac{ - 3± \sqrt{ {3}^{2} - 4 \times 1 \times 5 } }{2 \times 1} [/tex]
Any expression multiplied by 1 remains the same
[tex] \tt \: x = \frac{ - 3± \sqrt{ {3}^{2} - 4 \times 5 } }{2} [/tex]
Evaluate the power
[tex] \tt \: x = \frac{ - 3± \sqrt{9 - 4 \times 5} }{2} [/tex]
Multiply the numbers
[tex] \tt \: x = \frac{ - 3± \sqrt{9 - 20} }{2} [/tex]
Calculate the difference
[tex] \tt \: x = \frac{ - 3± \sqrt{ - 11} }{2} [/tex]
Calculate the square root
[tex] \tt \: x = \frac{ - 3± \sqrt{11}i }{2} [/tex]
Write the solutions, one with a + sign and one with a - sign
[tex] \tt \: x = \frac{ - 3 + \sqrt{11} i}{2} \\ \tt \: x = \frac{ - 3 - \sqrt{11}i }{2} [/tex]
Separate the real and the imaginary parts
[tex] \tt \: x = - \frac{3}{2} + \frac{ \sqrt{11} }{2} i \\ \tt \: x = - \frac{3}{2} - \frac{ \sqrt{11} }{2} i[/tex]
∴ The quadratic equation has two solutions
[tex] \rm x_{1} = - \frac{3}{2} + \frac{ \sqrt{11} }{2} i, x_{2} = - \frac{3}{2} - \frac{ \sqrt{11} }{2} i[/tex]
Answer:
[tex]⇒x = \frac{ - b \binom{ + }{ - } \sqrt{ {b}^{2} } - 4ac }{2a} [/tex]
Once in standard form, identify a, b and c from the original equation and plug them into the quadratic formula.
[tex]⇒ {x}^{2} + 3x + 5 = 0 \\ ⇒a = 1 \\ ⇒b = 3 \\ ⇒c = 5 \\ ⇒x = \frac{ - 3 \binom{ + }{ - } \sqrt{ {3}^{2} } - 4.1.5}{2.1} [/tex]
▪Evaluate the exponent
▪Multiply the numbers
▪Subtract the numbers
▪Multiply the numbers
[tex]⇒x = \frac{ - 3 \binom{ + }{ - } \sqrt{ - 11} }{2} [/tex]
▪The square root of a negative number is not a real number
[tex]⇒d = - 11[/tex]
♧ no solution ♧
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