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4x^{2}+4x-8=1
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.
4x^{2}+4x-8-1=1-1
Subtract 1 from both sides of the equation.
4x^{2}+4x-8-1=0
Subtracting 1 from itself leaves 0.
4x^{2}+4x-9=0
Subtract 1 from -8.
x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-9\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 4 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 4\left(-9\right)}}{2\times 4}
Square 4.
x=\frac{-4±\sqrt{16-16\left(-9\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-4±\sqrt{16+144}}{2\times 4}
Multiply -16 times -9.
x=\frac{-4±\sqrt{160}}{2\times 4}
Add 16 to 144.
x=\frac{-4±4\sqrt{10}}{2\times 4}
Take the square root of 160.
x=\frac{-4±4\sqrt{10}}{8}
Multiply 2 times 4.
x=\frac{4\sqrt{10}-4}{8}
Now solve the equation x=\frac{-4±4\sqrt{10}}{8} when ± is plus. Add -4 to 4\sqrt{10}.
x=\frac{\sqrt{10}-1}{2}
Divide -4+4\sqrt{10} by 8.
x=\frac{-4\sqrt{10}-4}{8}
Now solve the equation x=\frac{-4±4\sqrt{10}}{8} when ± is minus. Subtract 4\sqrt{10} from -4.
x=\frac{-\sqrt{10}-1}{2}
Divide -4-4\sqrt{10} by 8.
x=\frac{\sqrt{10}-1}{2} x=\frac{-\sqrt{10}-1}{2}
The equation is now solved.
4x^{2}+4x-8=1
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.
4x^{2}+4x-8-\left(-8\right)=1-\left(-8\right)
Add 8 to both sides of the equation.
4x^{2}+4x=1-\left(-8\right)
Subtracting -8 from itself leaves 0.
4x^{2}+4x=9
Subtract -8 from 1.
\frac{4x^{2}+4x}{4}=\frac{9}{4}
Divide both sides by 4.
x^{2}+\frac{4}{4}x=\frac{9}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+x=\frac{9}{4}
Divide 4 by 4.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{9}{4}+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+x+\frac{1}{4}=\frac{9+1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+x+\frac{1}{4}=\frac{5}{2}
Add \frac{9}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{2}\right)^{2}=\frac{5}{2}
Factor x^{2}+x+\frac{1}{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+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{2}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{\sqrt{10}}{2} x+\frac{1}{2}=-\frac{\sqrt{10}}{2}
Simplify.
x=\frac{\sqrt{10}-1}{2} x=\frac{-\sqrt{10}-1}{2}
Subtract \frac{1}{2} from both sides of the equation.