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x^{2}+4x-77=0
Subtract 77 from both sides.
a+b=4 ab=-77
To solve the equation, factor x^{2}+4x-77 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
-1,77 -7,11
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 -77.
-1+77=76 -7+11=4
Calculate the sum for each pair.
a=-7 b=11
The solution is the pair that gives sum 4.
\left(x-7\right)\left(x+11\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=7 x=-11
To find equation solutions, solve x-7=0 and x+11=0.
x^{2}+4x-77=0
Subtract 77 from both sides.
a+b=4 ab=1\left(-77\right)=-77
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-77. To find a and b, set up a system to be solved.
-1,77 -7,11
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 -77.
-1+77=76 -7+11=4
Calculate the sum for each pair.
a=-7 b=11
The solution is the pair that gives sum 4.
\left(x^{2}-7x\right)+\left(11x-77\right)
Rewrite x^{2}+4x-77 as \left(x^{2}-7x\right)+\left(11x-77\right).
x\left(x-7\right)+11\left(x-7\right)
Factor out x in the first and 11 in the second group.
\left(x-7\right)\left(x+11\right)
Factor out common term x-7 by using distributive property.
x=7 x=-11
To find equation solutions, solve x-7=0 and x+11=0.
x^{2}+4x=77
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^{2}+4x-77=77-77
Subtract 77 from both sides of the equation.
x^{2}+4x-77=0
Subtracting 77 from itself leaves 0.
x=\frac{-4±\sqrt{4^{2}-4\left(-77\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-77\right)}}{2}
Square 4.
x=\frac{-4±\sqrt{16+308}}{2}
Multiply -4 times -77.
x=\frac{-4±\sqrt{324}}{2}
Add 16 to 308.
x=\frac{-4±18}{2}
Take the square root of 324.
x=\frac{14}{2}
Now solve the equation x=\frac{-4±18}{2} when ± is plus. Add -4 to 18.
x=7
Divide 14 by 2.
x=-\frac{22}{2}
Now solve the equation x=\frac{-4±18}{2} when ± is minus. Subtract 18 from -4.
x=-11
Divide -22 by 2.
x=7 x=-11
The equation is now solved.
x^{2}+4x=77
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.
x^{2}+4x+2^{2}=77+2^{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=77+4
Square 2.
x^{2}+4x+4=81
Add 77 to 4.
\left(x+2\right)^{2}=81
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{81}
Take the square root of both sides of the equation.
x+2=9 x+2=-9
Simplify.
x=7 x=-11
Subtract 2 from both sides of the equation.