[ x ^ { 2 } - 16 x + 63 = 0 ]
Solve for x
x=7
x=9
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a+b=-16 ab=63
To solve the equation, factor x^{2}-16x+63 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,-63 -3,-21 -7,-9
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 63.
-1-63=-64 -3-21=-24 -7-9=-16
Calculate the sum for each pair.
a=-9 b=-7
The solution is the pair that gives sum -16.
\left(x-9\right)\left(x-7\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=9 x=7
To find equation solutions, solve x-9=0 and x-7=0.
a+b=-16 ab=1\times 63=63
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+63. To find a and b, set up a system to be solved.
-1,-63 -3,-21 -7,-9
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 63.
-1-63=-64 -3-21=-24 -7-9=-16
Calculate the sum for each pair.
a=-9 b=-7
The solution is the pair that gives sum -16.
\left(x^{2}-9x\right)+\left(-7x+63\right)
Rewrite x^{2}-16x+63 as \left(x^{2}-9x\right)+\left(-7x+63\right).
x\left(x-9\right)-7\left(x-9\right)
Factor out x in the first and -7 in the second group.
\left(x-9\right)\left(x-7\right)
Factor out common term x-9 by using distributive property.
x=9 x=7
To find equation solutions, solve x-9=0 and x-7=0.
x^{2}-16x+63=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{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 63}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-16\right)±\sqrt{256-4\times 63}}{2}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-252}}{2}
Multiply -4 times 63.
x=\frac{-\left(-16\right)±\sqrt{4}}{2}
Add 256 to -252.
x=\frac{-\left(-16\right)±2}{2}
Take the square root of 4.
x=\frac{16±2}{2}
The opposite of -16 is 16.
x=\frac{18}{2}
Now solve the equation x=\frac{16±2}{2} when ± is plus. Add 16 to 2.
x=9
Divide 18 by 2.
x=\frac{14}{2}
Now solve the equation x=\frac{16±2}{2} when ± is minus. Subtract 2 from 16.
x=7
Divide 14 by 2.
x=9 x=7
The equation is now solved.
x^{2}-16x+63=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.
x^{2}-16x+63-63=-63
Subtract 63 from both sides of the equation.
x^{2}-16x=-63
Subtracting 63 from itself leaves 0.
x^{2}-16x+\left(-8\right)^{2}=-63+\left(-8\right)^{2}
Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-16x+64=-63+64
Square -8.
x^{2}-16x+64=1
Add -63 to 64.
\left(x-8\right)^{2}=1
Factor x^{2}-16x+64. 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-8\right)^{2}}=\sqrt{1}
Take the square root of both sides of the equation.
x-8=1 x-8=-1
Simplify.
x=9 x=7
Add 8 to both sides of the equation.
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Limits
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