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6p^{2}+6p-7=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.
p=\frac{-6±\sqrt{6^{2}-4\times 6\left(-7\right)}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 6 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
p=\frac{-6±\sqrt{36-4\times 6\left(-7\right)}}{2\times 6}
Square 6.
p=\frac{-6±\sqrt{36-24\left(-7\right)}}{2\times 6}
Multiply -4 times 6.
p=\frac{-6±\sqrt{36+168}}{2\times 6}
Multiply -24 times -7.
p=\frac{-6±\sqrt{204}}{2\times 6}
Add 36 to 168.
p=\frac{-6±2\sqrt{51}}{2\times 6}
Take the square root of 204.
p=\frac{-6±2\sqrt{51}}{12}
Multiply 2 times 6.
p=\frac{2\sqrt{51}-6}{12}
Now solve the equation p=\frac{-6±2\sqrt{51}}{12} when ± is plus. Add -6 to 2\sqrt{51}.
p=\frac{\sqrt{51}}{6}-\frac{1}{2}
Divide -6+2\sqrt{51} by 12.
p=\frac{-2\sqrt{51}-6}{12}
Now solve the equation p=\frac{-6±2\sqrt{51}}{12} when ± is minus. Subtract 2\sqrt{51} from -6.
p=-\frac{\sqrt{51}}{6}-\frac{1}{2}
Divide -6-2\sqrt{51} by 12.
p=\frac{\sqrt{51}}{6}-\frac{1}{2} p=-\frac{\sqrt{51}}{6}-\frac{1}{2}
The equation is now solved.
6p^{2}+6p-7=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.
6p^{2}+6p-7-\left(-7\right)=-\left(-7\right)
Add 7 to both sides of the equation.
6p^{2}+6p=-\left(-7\right)
Subtracting -7 from itself leaves 0.
6p^{2}+6p=7
Subtract -7 from 0.
\frac{6p^{2}+6p}{6}=\frac{7}{6}
Divide both sides by 6.
p^{2}+\frac{6}{6}p=\frac{7}{6}
Dividing by 6 undoes the multiplication by 6.
p^{2}+p=\frac{7}{6}
Divide 6 by 6.
p^{2}+p+\left(\frac{1}{2}\right)^{2}=\frac{7}{6}+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
p^{2}+p+\frac{1}{4}=\frac{7}{6}+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
p^{2}+p+\frac{1}{4}=\frac{17}{12}
Add \frac{7}{6} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(p+\frac{1}{2}\right)^{2}=\frac{17}{12}
Factor p^{2}+p+\frac{1}{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(p+\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{12}}
Take the square root of both sides of the equation.
p+\frac{1}{2}=\frac{\sqrt{51}}{6} p+\frac{1}{2}=-\frac{\sqrt{51}}{6}
Simplify.
p=\frac{\sqrt{51}}{6}-\frac{1}{2} p=-\frac{\sqrt{51}}{6}-\frac{1}{2}
Subtract \frac{1}{2} from both sides of the equation.
x ^ 2 +1x -\frac{7}{6} = 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 6
r + s = -1 rs = -\frac{7}{6}
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}{2} - u s = -\frac{1}{2} + u
Two numbers r and s sum up to -1 exactly when the average of the two numbers is \frac{1}{2}*-1 = -\frac{1}{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>
(-\frac{1}{2} - u) (-\frac{1}{2} + u) = -\frac{7}{6}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{6}
\frac{1}{4} - u^2 = -\frac{7}{6}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{7}{6}-\frac{1}{4} = -\frac{17}{12}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{17}{12} u = \pm\sqrt{\frac{17}{12}} = \pm \frac{\sqrt{17}}{\sqrt{12}}
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}{2} - \frac{\sqrt{17}}{\sqrt{12}} = -1.690 s = -\frac{1}{2} + \frac{\sqrt{17}}{\sqrt{12}} = 0.690
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.