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a+b=8 ab=5\left(-4\right)=-20
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-4. To find a and b, set up a system to be solved.
-1,20 -2,10 -4,5
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.
-1+20=19 -2+10=8 -4+5=1
Calculate the sum for each pair.
a=-2 b=10
The solution is the pair that gives sum 8.
\left(5x^{2}-2x\right)+\left(10x-4\right)
Rewrite 5x^{2}+8x-4 as \left(5x^{2}-2x\right)+\left(10x-4\right).
x\left(5x-2\right)+2\left(5x-2\right)
Factor out x in the first and 2 in the second group.
\left(5x-2\right)\left(x+2\right)
Factor out common term 5x-2 by using distributive property.
x=\frac{2}{5} x=-2
To find equation solutions, solve 5x-2=0 and x+2=0.
5x^{2}+8x-4=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{-8±\sqrt{8^{2}-4\times 5\left(-4\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 8 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 5\left(-4\right)}}{2\times 5}
Square 8.
x=\frac{-8±\sqrt{64-20\left(-4\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-8±\sqrt{64+80}}{2\times 5}
Multiply -20 times -4.
x=\frac{-8±\sqrt{144}}{2\times 5}
Add 64 to 80.
x=\frac{-8±12}{2\times 5}
Take the square root of 144.
x=\frac{-8±12}{10}
Multiply 2 times 5.
x=\frac{4}{10}
Now solve the equation x=\frac{-8±12}{10} when ± is plus. Add -8 to 12.
x=\frac{2}{5}
Reduce the fraction \frac{4}{10} to lowest terms by extracting and canceling out 2.
x=-\frac{20}{10}
Now solve the equation x=\frac{-8±12}{10} when ± is minus. Subtract 12 from -8.
x=-2
Divide -20 by 10.
x=\frac{2}{5} x=-2
The equation is now solved.
5x^{2}+8x-4=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.
5x^{2}+8x-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
5x^{2}+8x=-\left(-4\right)
Subtracting -4 from itself leaves 0.
5x^{2}+8x=4
Subtract -4 from 0.
\frac{5x^{2}+8x}{5}=\frac{4}{5}
Divide both sides by 5.
x^{2}+\frac{8}{5}x=\frac{4}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+\frac{8}{5}x+\left(\frac{4}{5}\right)^{2}=\frac{4}{5}+\left(\frac{4}{5}\right)^{2}
Divide \frac{8}{5}, the coefficient of the x term, by 2 to get \frac{4}{5}. Then add the square of \frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{8}{5}x+\frac{16}{25}=\frac{4}{5}+\frac{16}{25}
Square \frac{4}{5} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{8}{5}x+\frac{16}{25}=\frac{36}{25}
Add \frac{4}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{4}{5}\right)^{2}=\frac{36}{25}
Factor x^{2}+\frac{8}{5}x+\frac{16}{25}. 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{4}{5}\right)^{2}}=\sqrt{\frac{36}{25}}
Take the square root of both sides of the equation.
x+\frac{4}{5}=\frac{6}{5} x+\frac{4}{5}=-\frac{6}{5}
Simplify.
x=\frac{2}{5} x=-2
Subtract \frac{4}{5} from both sides of the equation.