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2x^{2}-5+4x=0
Add 4x to both sides.
2x^{2}+4x-5=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{-4±\sqrt{4^{2}-4\times 2\left(-5\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 2\left(-5\right)}}{2\times 2}
Square 4.
x=\frac{-4±\sqrt{16-8\left(-5\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-4±\sqrt{16+40}}{2\times 2}
Multiply -8 times -5.
x=\frac{-4±\sqrt{56}}{2\times 2}
Add 16 to 40.
x=\frac{-4±2\sqrt{14}}{2\times 2}
Take the square root of 56.
x=\frac{-4±2\sqrt{14}}{4}
Multiply 2 times 2.
x=\frac{2\sqrt{14}-4}{4}
Now solve the equation x=\frac{-4±2\sqrt{14}}{4} when ± is plus. Add -4 to 2\sqrt{14}.
x=\frac{\sqrt{14}}{2}-1
Divide -4+2\sqrt{14} by 4.
x=\frac{-2\sqrt{14}-4}{4}
Now solve the equation x=\frac{-4±2\sqrt{14}}{4} when ± is minus. Subtract 2\sqrt{14} from -4.
x=-\frac{\sqrt{14}}{2}-1
Divide -4-2\sqrt{14} by 4.
x=\frac{\sqrt{14}}{2}-1 x=-\frac{\sqrt{14}}{2}-1
The equation is now solved.
2x^{2}-5+4x=0
Add 4x to both sides.
2x^{2}+4x=5
Add 5 to both sides. Anything plus zero gives itself.
\frac{2x^{2}+4x}{2}=\frac{5}{2}
Divide both sides by 2.
x^{2}+\frac{4}{2}x=\frac{5}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+2x=\frac{5}{2}
Divide 4 by 2.
x^{2}+2x+1^{2}=\frac{5}{2}+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=\frac{5}{2}+1
Square 1.
x^{2}+2x+1=\frac{7}{2}
Add \frac{5}{2} to 1.
\left(x+1\right)^{2}=\frac{7}{2}
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{\frac{7}{2}}
Take the square root of both sides of the equation.
x+1=\frac{\sqrt{14}}{2} x+1=-\frac{\sqrt{14}}{2}
Simplify.
x=\frac{\sqrt{14}}{2}-1 x=-\frac{\sqrt{14}}{2}-1
Subtract 1 from both sides of the equation.