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x^{2}+x-34+4=0
Add 4 to both sides.
x^{2}+x-30=0
Add -34 and 4 to get -30.
a+b=1 ab=-30
To solve the equation, factor x^{2}+x-30 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.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=-5 b=6
The solution is the pair that gives sum 1.
\left(x-5\right)\left(x+6\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=5 x=-6
To find equation solutions, solve x-5=0 and x+6=0.
x^{2}+x-34+4=0
Add 4 to both sides.
x^{2}+x-30=0
Add -34 and 4 to get -30.
a+b=1 ab=1\left(-30\right)=-30
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-30. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=-5 b=6
The solution is the pair that gives sum 1.
\left(x^{2}-5x\right)+\left(6x-30\right)
Rewrite x^{2}+x-30 as \left(x^{2}-5x\right)+\left(6x-30\right).
x\left(x-5\right)+6\left(x-5\right)
Factor out x in the first and 6 in the second group.
\left(x-5\right)\left(x+6\right)
Factor out common term x-5 by using distributive property.
x=5 x=-6
To find equation solutions, solve x-5=0 and x+6=0.
x^{2}+x-34=-4
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}+x-34-\left(-4\right)=-4-\left(-4\right)
Add 4 to both sides of the equation.
x^{2}+x-34-\left(-4\right)=0
Subtracting -4 from itself leaves 0.
x^{2}+x-30=0
Subtract -4 from -34.
x=\frac{-1±\sqrt{1^{2}-4\left(-30\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-30\right)}}{2}
Square 1.
x=\frac{-1±\sqrt{1+120}}{2}
Multiply -4 times -30.
x=\frac{-1±\sqrt{121}}{2}
Add 1 to 120.
x=\frac{-1±11}{2}
Take the square root of 121.
x=\frac{10}{2}
Now solve the equation x=\frac{-1±11}{2} when ± is plus. Add -1 to 11.
x=5
Divide 10 by 2.
x=-\frac{12}{2}
Now solve the equation x=\frac{-1±11}{2} when ± is minus. Subtract 11 from -1.
x=-6
Divide -12 by 2.
x=5 x=-6
The equation is now solved.
x^{2}+x-34=-4
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}+x-34-\left(-34\right)=-4-\left(-34\right)
Add 34 to both sides of the equation.
x^{2}+x=-4-\left(-34\right)
Subtracting -34 from itself leaves 0.
x^{2}+x=30
Subtract -34 from -4.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=30+\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}=30+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+x+\frac{1}{4}=\frac{121}{4}
Add 30 to \frac{1}{4}.
\left(x+\frac{1}{2}\right)^{2}=\frac{121}{4}
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{121}{4}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{11}{2} x+\frac{1}{2}=-\frac{11}{2}
Simplify.
x=5 x=-6
Subtract \frac{1}{2} from both sides of the equation.