Solve for x
x=-7
x=-3
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a+b=-10 ab=-\left(-21\right)=21
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-21. To find a and b, set up a system to be solved.
-1,-21 -3,-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 21.
-1-21=-22 -3-7=-10
Calculate the sum for each pair.
a=-3 b=-7
The solution is the pair that gives sum -10.
\left(-x^{2}-3x\right)+\left(-7x-21\right)
Rewrite -x^{2}-10x-21 as \left(-x^{2}-3x\right)+\left(-7x-21\right).
x\left(-x-3\right)+7\left(-x-3\right)
Factor out x in the first and 7 in the second group.
\left(-x-3\right)\left(x+7\right)
Factor out common term -x-3 by using distributive property.
x=-3 x=-7
To find equation solutions, solve -x-3=0 and x+7=0.
-x^{2}-10x-21=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(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -10 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+4\left(-21\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-10\right)±\sqrt{100-84}}{2\left(-1\right)}
Multiply 4 times -21.
x=\frac{-\left(-10\right)±\sqrt{16}}{2\left(-1\right)}
Add 100 to -84.
x=\frac{-\left(-10\right)±4}{2\left(-1\right)}
Take the square root of 16.
x=\frac{10±4}{2\left(-1\right)}
The opposite of -10 is 10.
x=\frac{10±4}{-2}
Multiply 2 times -1.
x=\frac{14}{-2}
Now solve the equation x=\frac{10±4}{-2} when ± is plus. Add 10 to 4.
x=-7
Divide 14 by -2.
x=\frac{6}{-2}
Now solve the equation x=\frac{10±4}{-2} when ± is minus. Subtract 4 from 10.
x=-3
Divide 6 by -2.
x=-7 x=-3
The equation is now solved.
-x^{2}-10x-21=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}-10x-21-\left(-21\right)=-\left(-21\right)
Add 21 to both sides of the equation.
-x^{2}-10x=-\left(-21\right)
Subtracting -21 from itself leaves 0.
-x^{2}-10x=21
Subtract -21 from 0.
\frac{-x^{2}-10x}{-1}=\frac{21}{-1}
Divide both sides by -1.
x^{2}+\left(-\frac{10}{-1}\right)x=\frac{21}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}+10x=\frac{21}{-1}
Divide -10 by -1.
x^{2}+10x=-21
Divide 21 by -1.
x^{2}+10x+5^{2}=-21+5^{2}
Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+10x+25=-21+25
Square 5.
x^{2}+10x+25=4
Add -21 to 25.
\left(x+5\right)^{2}=4
Factor x^{2}+10x+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+5\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
x+5=2 x+5=-2
Simplify.
x=-3 x=-7
Subtract 5 from both sides of the equation.
x ^ 2 +10x +21 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = -10 rs = 21
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -5 - u s = -5 + u
Two numbers r and s sum up to -10 exactly when the average of the two numbers is \frac{1}{2}*-10 = -5. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-5 - u) (-5 + u) = 21
To solve for unknown quantity u, substitute these in the product equation rs = 21
25 - u^2 = 21
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 21-25 = -4
Simplify the expression by subtracting 25 on both sides
u^2 = 4 u = \pm\sqrt{4} = \pm 2
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-5 - 2 = -7 s = -5 + 2 = -3
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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