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2x^{2}-8x-3=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-3\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -8 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-3\right)}}{2\times 2}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64-8\left(-3\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-8\right)±\sqrt{64+24}}{2\times 2}
Multiply -8 times -3.
x=\frac{-\left(-8\right)±\sqrt{88}}{2\times 2}
Add 64 to 24.
x=\frac{-\left(-8\right)±2\sqrt{22}}{2\times 2}
Take the square root of 88.
x=\frac{8±2\sqrt{22}}{2\times 2}
The opposite of -8 is 8.
x=\frac{8±2\sqrt{22}}{4}
Multiply 2 times 2.
x=\frac{2\sqrt{22}+8}{4}
Now solve the equation x=\frac{8±2\sqrt{22}}{4} when ± is plus. Add 8 to 2\sqrt{22}.
x=\frac{\sqrt{22}}{2}+2
Divide 8+2\sqrt{22} by 4.
x=\frac{8-2\sqrt{22}}{4}
Now solve the equation x=\frac{8±2\sqrt{22}}{4} when ± is minus. Subtract 2\sqrt{22} from 8.
x=-\frac{\sqrt{22}}{2}+2
Divide 8-2\sqrt{22} by 4.
x=\frac{\sqrt{22}}{2}+2 x=-\frac{\sqrt{22}}{2}+2
The equation is now solved.
2x^{2}-8x-3=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
2x^{2}-8x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
2x^{2}-8x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
2x^{2}-8x=3
Subtract -3 from 0.
\frac{2x^{2}-8x}{2}=\frac{3}{2}
Divide both sides by 2.
x^{2}+\left(-\frac{8}{2}\right)x=\frac{3}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-4x=\frac{3}{2}
Divide -8 by 2.
x^{2}-4x+\left(-2\right)^{2}=\frac{3}{2}+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=\frac{3}{2}+4
Square -2.
x^{2}-4x+4=\frac{11}{2}
Add \frac{3}{2} to 4.
\left(x-2\right)^{2}=\frac{11}{2}
Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{11}{2}}
Take the square root of both sides of the equation.
x-2=\frac{\sqrt{22}}{2} x-2=-\frac{\sqrt{22}}{2}
Simplify.
x=\frac{\sqrt{22}}{2}+2 x=-\frac{\sqrt{22}}{2}+2
Add 2 to both sides of the equation.