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4x^{2}-10x-37=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{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 4\left(-37\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -10 for b, and -37 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\times 4\left(-37\right)}}{2\times 4}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100-16\left(-37\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-10\right)±\sqrt{100+592}}{2\times 4}
Multiply -16 times -37.
x=\frac{-\left(-10\right)±\sqrt{692}}{2\times 4}
Add 100 to 592.
x=\frac{-\left(-10\right)±2\sqrt{173}}{2\times 4}
Take the square root of 692.
x=\frac{10±2\sqrt{173}}{2\times 4}
The opposite of -10 is 10.
x=\frac{10±2\sqrt{173}}{8}
Multiply 2 times 4.
x=\frac{2\sqrt{173}+10}{8}
Now solve the equation x=\frac{10±2\sqrt{173}}{8} when ± is plus. Add 10 to 2\sqrt{173}.
x=\frac{\sqrt{173}+5}{4}
Divide 10+2\sqrt{173} by 8.
x=\frac{10-2\sqrt{173}}{8}
Now solve the equation x=\frac{10±2\sqrt{173}}{8} when ± is minus. Subtract 2\sqrt{173} from 10.
x=\frac{5-\sqrt{173}}{4}
Divide 10-2\sqrt{173} by 8.
x=\frac{\sqrt{173}+5}{4} x=\frac{5-\sqrt{173}}{4}
The equation is now solved.
4x^{2}-10x-37=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}-10x-37-\left(-37\right)=-\left(-37\right)
Add 37 to both sides of the equation.
4x^{2}-10x=-\left(-37\right)
Subtracting -37 from itself leaves 0.
4x^{2}-10x=37
Subtract -37 from 0.
\frac{4x^{2}-10x}{4}=\frac{37}{4}
Divide both sides by 4.
x^{2}+\left(-\frac{10}{4}\right)x=\frac{37}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{5}{2}x=\frac{37}{4}
Reduce the fraction \frac{-10}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=\frac{37}{4}+\left(-\frac{5}{4}\right)^{2}
Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{37}{4}+\frac{25}{16}
Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{173}{16}
Add \frac{37}{4} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{5}{4}\right)^{2}=\frac{173}{16}
Factor x^{2}-\frac{5}{2}x+\frac{25}{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{5}{4}\right)^{2}}=\sqrt{\frac{173}{16}}
Take the square root of both sides of the equation.
x-\frac{5}{4}=\frac{\sqrt{173}}{4} x-\frac{5}{4}=-\frac{\sqrt{173}}{4}
Simplify.
x=\frac{\sqrt{173}+5}{4} x=\frac{5-\sqrt{173}}{4}
Add \frac{5}{4} to both sides of the equation.
x ^ 2 -\frac{5}{2}x -\frac{37}{4} = 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{5}{2} rs = -\frac{37}{4}
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{5}{4} - u s = \frac{5}{4} + u
Two numbers r and s sum up to \frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{2} = \frac{5}{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{5}{4} - u) (\frac{5}{4} + u) = -\frac{37}{4}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{37}{4}
\frac{25}{16} - u^2 = -\frac{37}{4}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{37}{4}-\frac{25}{16} = -\frac{173}{16}
Simplify the expression by subtracting \frac{25}{16} on both sides
u^2 = \frac{173}{16} u = \pm\sqrt{\frac{173}{16}} = \pm \frac{\sqrt{173}}{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{5}{4} - \frac{\sqrt{173}}{4} = -2.038 s = \frac{5}{4} + \frac{\sqrt{173}}{4} = 4.538
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.