Solve for z
z = \frac{\sqrt{241} + 1}{10} \approx 1.65241747
z=\frac{1-\sqrt{241}}{10}\approx -1.45241747
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-5z^{2}+z+12=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.
z=\frac{-1±\sqrt{1^{2}-4\left(-5\right)\times 12}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
z=\frac{-1±\sqrt{1-4\left(-5\right)\times 12}}{2\left(-5\right)}
Square 1.
z=\frac{-1±\sqrt{1+20\times 12}}{2\left(-5\right)}
Multiply -4 times -5.
z=\frac{-1±\sqrt{1+240}}{2\left(-5\right)}
Multiply 20 times 12.
z=\frac{-1±\sqrt{241}}{2\left(-5\right)}
Add 1 to 240.
z=\frac{-1±\sqrt{241}}{-10}
Multiply 2 times -5.
z=\frac{\sqrt{241}-1}{-10}
Now solve the equation z=\frac{-1±\sqrt{241}}{-10} when ± is plus. Add -1 to \sqrt{241}.
z=\frac{1-\sqrt{241}}{10}
Divide -1+\sqrt{241} by -10.
z=\frac{-\sqrt{241}-1}{-10}
Now solve the equation z=\frac{-1±\sqrt{241}}{-10} when ± is minus. Subtract \sqrt{241} from -1.
z=\frac{\sqrt{241}+1}{10}
Divide -1-\sqrt{241} by -10.
z=\frac{1-\sqrt{241}}{10} z=\frac{\sqrt{241}+1}{10}
The equation is now solved.
-5z^{2}+z+12=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.
-5z^{2}+z+12-12=-12
Subtract 12 from both sides of the equation.
-5z^{2}+z=-12
Subtracting 12 from itself leaves 0.
\frac{-5z^{2}+z}{-5}=-\frac{12}{-5}
Divide both sides by -5.
z^{2}+\frac{1}{-5}z=-\frac{12}{-5}
Dividing by -5 undoes the multiplication by -5.
z^{2}-\frac{1}{5}z=-\frac{12}{-5}
Divide 1 by -5.
z^{2}-\frac{1}{5}z=\frac{12}{5}
Divide -12 by -5.
z^{2}-\frac{1}{5}z+\left(-\frac{1}{10}\right)^{2}=\frac{12}{5}+\left(-\frac{1}{10}\right)^{2}
Divide -\frac{1}{5}, the coefficient of the x term, by 2 to get -\frac{1}{10}. Then add the square of -\frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
z^{2}-\frac{1}{5}z+\frac{1}{100}=\frac{12}{5}+\frac{1}{100}
Square -\frac{1}{10} by squaring both the numerator and the denominator of the fraction.
z^{2}-\frac{1}{5}z+\frac{1}{100}=\frac{241}{100}
Add \frac{12}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(z-\frac{1}{10}\right)^{2}=\frac{241}{100}
Factor z^{2}-\frac{1}{5}z+\frac{1}{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(z-\frac{1}{10}\right)^{2}}=\sqrt{\frac{241}{100}}
Take the square root of both sides of the equation.
z-\frac{1}{10}=\frac{\sqrt{241}}{10} z-\frac{1}{10}=-\frac{\sqrt{241}}{10}
Simplify.
z=\frac{\sqrt{241}+1}{10} z=\frac{1-\sqrt{241}}{10}
Add \frac{1}{10} to both sides of the equation.
x ^ 2 -\frac{1}{5}x -\frac{12}{5} = 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 = \frac{1}{5} rs = -\frac{12}{5}
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{1}{10} - u s = \frac{1}{10} + u
Two numbers r and s sum up to \frac{1}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{5} = \frac{1}{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>
(\frac{1}{10} - u) (\frac{1}{10} + u) = -\frac{12}{5}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{12}{5}
\frac{1}{100} - u^2 = -\frac{12}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{12}{5}-\frac{1}{100} = -\frac{241}{100}
Simplify the expression by subtracting \frac{1}{100} on both sides
u^2 = \frac{241}{100} u = \pm\sqrt{\frac{241}{100}} = \pm \frac{\sqrt{241}}{10}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{1}{10} - \frac{\sqrt{241}}{10} = -1.452 s = \frac{1}{10} + \frac{\sqrt{241}}{10} = 1.652
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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