Solve for r
r=10
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-7r^{2}+140r-700=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.
r=\frac{-140±\sqrt{140^{2}-4\left(-7\right)\left(-700\right)}}{2\left(-7\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 140 for b, and -700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
r=\frac{-140±\sqrt{19600-4\left(-7\right)\left(-700\right)}}{2\left(-7\right)}
Square 140.
r=\frac{-140±\sqrt{19600+28\left(-700\right)}}{2\left(-7\right)}
Multiply -4 times -7.
r=\frac{-140±\sqrt{19600-19600}}{2\left(-7\right)}
Multiply 28 times -700.
r=\frac{-140±\sqrt{0}}{2\left(-7\right)}
Add 19600 to -19600.
r=-\frac{140}{2\left(-7\right)}
Take the square root of 0.
r=-\frac{140}{-14}
Multiply 2 times -7.
r=10
Divide -140 by -14.
-7r^{2}+140r-700=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.
-7r^{2}+140r-700-\left(-700\right)=-\left(-700\right)
Add 700 to both sides of the equation.
-7r^{2}+140r=-\left(-700\right)
Subtracting -700 from itself leaves 0.
-7r^{2}+140r=700
Subtract -700 from 0.
\frac{-7r^{2}+140r}{-7}=\frac{700}{-7}
Divide both sides by -7.
r^{2}+\frac{140}{-7}r=\frac{700}{-7}
Dividing by -7 undoes the multiplication by -7.
r^{2}-20r=\frac{700}{-7}
Divide 140 by -7.
r^{2}-20r=-100
Divide 700 by -7.
r^{2}-20r+\left(-10\right)^{2}=-100+\left(-10\right)^{2}
Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
r^{2}-20r+100=-100+100
Square -10.
r^{2}-20r+100=0
Add -100 to 100.
\left(r-10\right)^{2}=0
Factor r^{2}-20r+100. 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(r-10\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
r-10=0 r-10=0
Simplify.
r=10 r=10
Add 10 to both sides of the equation.
r=10
The equation is now solved. Solutions are the same.
x ^ 2 -20x +100 = 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 = 20 rs = 100
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 = 10 - u s = 10 + u
Two numbers r and s sum up to 20 exactly when the average of the two numbers is \frac{1}{2}*20 = 10. 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>
(10 - u) (10 + u) = 100
To solve for unknown quantity u, substitute these in the product equation rs = 100
100 - u^2 = 100
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 100-100 = 0
Simplify the expression by subtracting 100 on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r = s = 10
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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