Solve for x
x=2\sqrt{37}-2\approx 10.165525061
x=-2\sqrt{37}-2\approx -14.165525061
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5x^{2}+20x-720=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{-20±\sqrt{20^{2}-4\times 5\left(-720\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 20 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-20±\sqrt{400-4\times 5\left(-720\right)}}{2\times 5}
Square 20.
x=\frac{-20±\sqrt{400-20\left(-720\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-20±\sqrt{400+14400}}{2\times 5}
Multiply -20 times -720.
x=\frac{-20±\sqrt{14800}}{2\times 5}
Add 400 to 14400.
x=\frac{-20±20\sqrt{37}}{2\times 5}
Take the square root of 14800.
x=\frac{-20±20\sqrt{37}}{10}
Multiply 2 times 5.
x=\frac{20\sqrt{37}-20}{10}
Now solve the equation x=\frac{-20±20\sqrt{37}}{10} when ± is plus. Add -20 to 20\sqrt{37}.
x=2\sqrt{37}-2
Divide -20+20\sqrt{37} by 10.
x=\frac{-20\sqrt{37}-20}{10}
Now solve the equation x=\frac{-20±20\sqrt{37}}{10} when ± is minus. Subtract 20\sqrt{37} from -20.
x=-2\sqrt{37}-2
Divide -20-20\sqrt{37} by 10.
x=2\sqrt{37}-2 x=-2\sqrt{37}-2
The equation is now solved.
5x^{2}+20x-720=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.
5x^{2}+20x-720-\left(-720\right)=-\left(-720\right)
Add 720 to both sides of the equation.
5x^{2}+20x=-\left(-720\right)
Subtracting -720 from itself leaves 0.
5x^{2}+20x=720
Subtract -720 from 0.
\frac{5x^{2}+20x}{5}=\frac{720}{5}
Divide both sides by 5.
x^{2}+\frac{20}{5}x=\frac{720}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+4x=\frac{720}{5}
Divide 20 by 5.
x^{2}+4x=144
Divide 720 by 5.
x^{2}+4x+2^{2}=144+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=144+4
Square 2.
x^{2}+4x+4=148
Add 144 to 4.
\left(x+2\right)^{2}=148
Factor x^{2}+4x+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+2\right)^{2}}=\sqrt{148}
Take the square root of both sides of the equation.
x+2=2\sqrt{37} x+2=-2\sqrt{37}
Simplify.
x=2\sqrt{37}-2 x=-2\sqrt{37}-2
Subtract 2 from both sides of the equation.
x ^ 2 +4x -144 = 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.This is achieved by dividing both sides of the equation by 5
r + s = -4 rs = -144
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 = -2 - u s = -2 + u
Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -2. 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>
(-2 - u) (-2 + u) = -144
To solve for unknown quantity u, substitute these in the product equation rs = -144
4 - u^2 = -144
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -144-4 = -148
Simplify the expression by subtracting 4 on both sides
u^2 = 148 u = \pm\sqrt{148} = \pm \sqrt{148}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-2 - \sqrt{148} = -14.166 s = -2 + \sqrt{148} = 10.166
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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Matrix
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Simultaneous equation
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Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
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Limits
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