Solve for x
x = \frac{\sqrt{329} - 13}{4} \approx 1.284589287
x=\frac{-\sqrt{329}-13}{4}\approx -7.784589287
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4x^{2}+26x-40=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{-26±\sqrt{26^{2}-4\times 4\left(-40\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 26 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-26±\sqrt{676-4\times 4\left(-40\right)}}{2\times 4}
Square 26.
x=\frac{-26±\sqrt{676-16\left(-40\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-26±\sqrt{676+640}}{2\times 4}
Multiply -16 times -40.
x=\frac{-26±\sqrt{1316}}{2\times 4}
Add 676 to 640.
x=\frac{-26±2\sqrt{329}}{2\times 4}
Take the square root of 1316.
x=\frac{-26±2\sqrt{329}}{8}
Multiply 2 times 4.
x=\frac{2\sqrt{329}-26}{8}
Now solve the equation x=\frac{-26±2\sqrt{329}}{8} when ± is plus. Add -26 to 2\sqrt{329}.
x=\frac{\sqrt{329}-13}{4}
Divide -26+2\sqrt{329} by 8.
x=\frac{-2\sqrt{329}-26}{8}
Now solve the equation x=\frac{-26±2\sqrt{329}}{8} when ± is minus. Subtract 2\sqrt{329} from -26.
x=\frac{-\sqrt{329}-13}{4}
Divide -26-2\sqrt{329} by 8.
x=\frac{\sqrt{329}-13}{4} x=\frac{-\sqrt{329}-13}{4}
The equation is now solved.
4x^{2}+26x-40=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.
4x^{2}+26x-40-\left(-40\right)=-\left(-40\right)
Add 40 to both sides of the equation.
4x^{2}+26x=-\left(-40\right)
Subtracting -40 from itself leaves 0.
4x^{2}+26x=40
Subtract -40 from 0.
\frac{4x^{2}+26x}{4}=\frac{40}{4}
Divide both sides by 4.
x^{2}+\frac{26}{4}x=\frac{40}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+\frac{13}{2}x=\frac{40}{4}
Reduce the fraction \frac{26}{4} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{13}{2}x=10
Divide 40 by 4.
x^{2}+\frac{13}{2}x+\left(\frac{13}{4}\right)^{2}=10+\left(\frac{13}{4}\right)^{2}
Divide \frac{13}{2}, the coefficient of the x term, by 2 to get \frac{13}{4}. Then add the square of \frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{13}{2}x+\frac{169}{16}=10+\frac{169}{16}
Square \frac{13}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{13}{2}x+\frac{169}{16}=\frac{329}{16}
Add 10 to \frac{169}{16}.
\left(x+\frac{13}{4}\right)^{2}=\frac{329}{16}
Factor x^{2}+\frac{13}{2}x+\frac{169}{16}. 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{13}{4}\right)^{2}}=\sqrt{\frac{329}{16}}
Take the square root of both sides of the equation.
x+\frac{13}{4}=\frac{\sqrt{329}}{4} x+\frac{13}{4}=-\frac{\sqrt{329}}{4}
Simplify.
x=\frac{\sqrt{329}-13}{4} x=\frac{-\sqrt{329}-13}{4}
Subtract \frac{13}{4} from both sides of the equation.
x ^ 2 +\frac{13}{2}x -10 = 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 4
r + s = -\frac{13}{2} rs = -10
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 = -\frac{13}{4} - u s = -\frac{13}{4} + u
Two numbers r and s sum up to -\frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{2} = -\frac{13}{4}. 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>
(-\frac{13}{4} - u) (-\frac{13}{4} + u) = -10
To solve for unknown quantity u, substitute these in the product equation rs = -10
\frac{169}{16} - u^2 = -10
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -10-\frac{169}{16} = -\frac{329}{16}
Simplify the expression by subtracting \frac{169}{16} on both sides
u^2 = \frac{329}{16} u = \pm\sqrt{\frac{329}{16}} = \pm \frac{\sqrt{329}}{4}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{13}{4} - \frac{\sqrt{329}}{4} = -7.785 s = -\frac{13}{4} + \frac{\sqrt{329}}{4} = 1.285
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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