[tex]2x^2 -5x-8 =0\\\\\implies x = \dfrac{ -(-5) \pm \sqrt{(-5)^2 - 4\cdot 2 \cdot (-8)}}{2 \cdot 2}\\\\\\\implies x = \dfrac{5 \pm \sqrt{89}}{4}\\\\\\\text{Hence}~ x = \dfrac{5 + \sqrt{89}}{4},~~ x = \dfrac{5 - \sqrt{89}}{4}[/tex]
Answer:
[tex]\sf \boxed{\sf x=\frac{5+\sqrt{89}}{4}}[/tex]
[tex]\sf \boxed{\sf x=\frac{5-\sqrt{89}}{4}}[/tex]
[tex]\sf 2x^2-5x-8=0[/tex]
☁ We'll solve this equation using The Quadratic Formula:
[tex]\boxed{\sf \cfrac{-b\pm \sqrt{b^2-4ac}}{2a}}[/tex]
A= 2, b= -5, c= -8
[tex]\sf x=\cfrac{-\left(-5\right)\pm \sqrt{\left(-5\right)^2-4\times \:2\left(-8\right)}}{2\times \:2}[/tex]
[tex]\mapsto \sf \sqrt{\left(-5\right)^2-4\times \:2\left(-8\right)}[/tex]
[tex]**\sf \left(-5\right)^2=5^2[/tex]
[tex]\mapsto \sf \sqrt{5^2+4\times \:2\times \:8}[/tex]
[tex]\mapsto \sf \sqrt{5^2+64}[/tex]
[tex]\mapsto \sf \sqrt{89}[/tex]
______________
[tex]\sf x=\cfrac{-\left(-5\right)\pm \sqrt{89}}{2\times \:2}[/tex]
☁ Now, we'll Separate solutions, first, we'll solve the equation when ± is plus:
[tex]\mapsto \sf \cfrac{-\left(-5\right)+\sqrt{89}}{2\times \:2}[/tex]
☁ Multiply 2*2 =4
[tex]\boxed{\sf \cfrac{5+\sqrt{89}}{4}}[/tex]
_________________
☁ Now solve the equation when ± is minus:
[tex]\mapsto \sf \cfrac{-\left(-5\right)-\sqrt{89}}{2\times \:2}[/tex]
☁ Multiply 2*2 =4
[tex]\boxed{\sf \cfrac{5-\sqrt{89}}{4}}[/tex]
[tex]\sf Solution \:1:\boxed{\sf x=\frac{5+\sqrt{89}}{4}}[/tex]
[tex]\sf Solution \:2:\boxed{\sf x=\frac{5-\sqrt{89}}{4}}[/tex]
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