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x^{2}-11x+28=0
Add 28 to both sides.
a+b=-11 ab=28
To solve the equation, factor x^{2}-11x+28 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,-28 -2,-14 -4,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.
-1-28=-29 -2-14=-16 -4-7=-11
Calculate the sum for each pair.
a=-7 b=-4
The solution is the pair that gives sum -11.
\left(x-7\right)\left(x-4\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=7 x=4
To find equation solutions, solve x-7=0 and x-4=0.
x^{2}-11x+28=0
Add 28 to both sides.
a+b=-11 ab=1\times 28=28
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.
-1,-28 -2,-14 -4,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.
-1-28=-29 -2-14=-16 -4-7=-11
Calculate the sum for each pair.
a=-7 b=-4
The solution is the pair that gives sum -11.
\left(x^{2}-7x\right)+\left(-4x+28\right)
Rewrite x^{2}-11x+28 as \left(x^{2}-7x\right)+\left(-4x+28\right).
x\left(x-7\right)-4\left(x-7\right)
Factor out x in the first and -4 in the second group.
\left(x-7\right)\left(x-4\right)
Factor out common term x-7 by using distributive property.
x=7 x=4
To find equation solutions, solve x-7=0 and x-4=0.
x^{2}-11x=-28
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}-11x-\left(-28\right)=-28-\left(-28\right)
Add 28 to both sides of the equation.
x^{2}-11x-\left(-28\right)=0
Subtracting -28 from itself leaves 0.
x^{2}-11x+28=0
Subtract -28 from 0.
x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 28}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-11\right)±\sqrt{121-4\times 28}}{2}
Square -11.
x=\frac{-\left(-11\right)±\sqrt{121-112}}{2}
Multiply -4 times 28.
x=\frac{-\left(-11\right)±\sqrt{9}}{2}
Add 121 to -112.
x=\frac{-\left(-11\right)±3}{2}
Take the square root of 9.
x=\frac{11±3}{2}
The opposite of -11 is 11.
x=\frac{14}{2}
Now solve the equation x=\frac{11±3}{2} when ± is plus. Add 11 to 3.
x=7
Divide 14 by 2.
x=\frac{8}{2}
Now solve the equation x=\frac{11±3}{2} when ± is minus. Subtract 3 from 11.
x=4
Divide 8 by 2.
x=7 x=4
The equation is now solved.
x^{2}-11x=-28
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}-11x+\left(-\frac{11}{2}\right)^{2}=-28+\left(-\frac{11}{2}\right)^{2}
Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-11x+\frac{121}{4}=-28+\frac{121}{4}
Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-11x+\frac{121}{4}=\frac{9}{4}
Add -28 to \frac{121}{4}.
\left(x-\frac{11}{2}\right)^{2}=\frac{9}{4}
Factor x^{2}-11x+\frac{121}{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{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}
Take the square root of both sides of the equation.
x-\frac{11}{2}=\frac{3}{2} x-\frac{11}{2}=-\frac{3}{2}
Simplify.
x=7 x=4
Add \frac{11}{2} to both sides of the equation.