Solve for q
q=\frac{\sqrt{21}-1}{5}\approx 0.716515139
q=\frac{-\sqrt{21}-1}{5}\approx -1.116515139
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5q^{2}+2q-4=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.
q=\frac{-2±\sqrt{2^{2}-4\times 5\left(-4\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 2 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
q=\frac{-2±\sqrt{4-4\times 5\left(-4\right)}}{2\times 5}
Square 2.
q=\frac{-2±\sqrt{4-20\left(-4\right)}}{2\times 5}
Multiply -4 times 5.
q=\frac{-2±\sqrt{4+80}}{2\times 5}
Multiply -20 times -4.
q=\frac{-2±\sqrt{84}}{2\times 5}
Add 4 to 80.
q=\frac{-2±2\sqrt{21}}{2\times 5}
Take the square root of 84.
q=\frac{-2±2\sqrt{21}}{10}
Multiply 2 times 5.
q=\frac{2\sqrt{21}-2}{10}
Now solve the equation q=\frac{-2±2\sqrt{21}}{10} when ± is plus. Add -2 to 2\sqrt{21}.
q=\frac{\sqrt{21}-1}{5}
Divide -2+2\sqrt{21} by 10.
q=\frac{-2\sqrt{21}-2}{10}
Now solve the equation q=\frac{-2±2\sqrt{21}}{10} when ± is minus. Subtract 2\sqrt{21} from -2.
q=\frac{-\sqrt{21}-1}{5}
Divide -2-2\sqrt{21} by 10.
q=\frac{\sqrt{21}-1}{5} q=\frac{-\sqrt{21}-1}{5}
The equation is now solved.
5q^{2}+2q-4=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.
5q^{2}+2q-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
5q^{2}+2q=-\left(-4\right)
Subtracting -4 from itself leaves 0.
5q^{2}+2q=4
Subtract -4 from 0.
\frac{5q^{2}+2q}{5}=\frac{4}{5}
Divide both sides by 5.
q^{2}+\frac{2}{5}q=\frac{4}{5}
Dividing by 5 undoes the multiplication by 5.
q^{2}+\frac{2}{5}q+\left(\frac{1}{5}\right)^{2}=\frac{4}{5}+\left(\frac{1}{5}\right)^{2}
Divide \frac{2}{5}, the coefficient of the x term, by 2 to get \frac{1}{5}. Then add the square of \frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
q^{2}+\frac{2}{5}q+\frac{1}{25}=\frac{4}{5}+\frac{1}{25}
Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.
q^{2}+\frac{2}{5}q+\frac{1}{25}=\frac{21}{25}
Add \frac{4}{5} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(q+\frac{1}{5}\right)^{2}=\frac{21}{25}
Factor q^{2}+\frac{2}{5}q+\frac{1}{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(q+\frac{1}{5}\right)^{2}}=\sqrt{\frac{21}{25}}
Take the square root of both sides of the equation.
q+\frac{1}{5}=\frac{\sqrt{21}}{5} q+\frac{1}{5}=-\frac{\sqrt{21}}{5}
Simplify.
q=\frac{\sqrt{21}-1}{5} q=\frac{-\sqrt{21}-1}{5}
Subtract \frac{1}{5} from both sides of the equation.
x ^ 2 +\frac{2}{5}x -\frac{4}{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.This is achieved by dividing both sides of the equation by 5
r + s = -\frac{2}{5} rs = -\frac{4}{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}{5} - u s = -\frac{1}{5} + u
Two numbers r and s sum up to -\frac{2}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{5} = -\frac{1}{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>
(-\frac{1}{5} - u) (-\frac{1}{5} + u) = -\frac{4}{5}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{5}
\frac{1}{25} - u^2 = -\frac{4}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{5}-\frac{1}{25} = -\frac{21}{25}
Simplify the expression by subtracting \frac{1}{25} on both sides
u^2 = \frac{21}{25} u = \pm\sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}
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}{5} - \frac{\sqrt{21}}{5} = -1.117 s = -\frac{1}{5} + \frac{\sqrt{21}}{5} = 0.717
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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