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-16t^{2}+14t+6=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.
t=\frac{-14±\sqrt{14^{2}-4\left(-16\right)\times 6}}{2\left(-16\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 14 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-14±\sqrt{196-4\left(-16\right)\times 6}}{2\left(-16\right)}
Square 14.
t=\frac{-14±\sqrt{196+64\times 6}}{2\left(-16\right)}
Multiply -4 times -16.
t=\frac{-14±\sqrt{196+384}}{2\left(-16\right)}
Multiply 64 times 6.
t=\frac{-14±\sqrt{580}}{2\left(-16\right)}
Add 196 to 384.
t=\frac{-14±2\sqrt{145}}{2\left(-16\right)}
Take the square root of 580.
t=\frac{-14±2\sqrt{145}}{-32}
Multiply 2 times -16.
t=\frac{2\sqrt{145}-14}{-32}
Now solve the equation t=\frac{-14±2\sqrt{145}}{-32} when ± is plus. Add -14 to 2\sqrt{145}.
t=\frac{7-\sqrt{145}}{16}
Divide -14+2\sqrt{145} by -32.
t=\frac{-2\sqrt{145}-14}{-32}
Now solve the equation t=\frac{-14±2\sqrt{145}}{-32} when ± is minus. Subtract 2\sqrt{145} from -14.
t=\frac{\sqrt{145}+7}{16}
Divide -14-2\sqrt{145} by -32.
t=\frac{7-\sqrt{145}}{16} t=\frac{\sqrt{145}+7}{16}
The equation is now solved.
-16t^{2}+14t+6=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.
-16t^{2}+14t+6-6=-6
Subtract 6 from both sides of the equation.
-16t^{2}+14t=-6
Subtracting 6 from itself leaves 0.
\frac{-16t^{2}+14t}{-16}=-\frac{6}{-16}
Divide both sides by -16.
t^{2}+\frac{14}{-16}t=-\frac{6}{-16}
Dividing by -16 undoes the multiplication by -16.
t^{2}-\frac{7}{8}t=-\frac{6}{-16}
Reduce the fraction \frac{14}{-16} to lowest terms by extracting and canceling out 2.
t^{2}-\frac{7}{8}t=\frac{3}{8}
Reduce the fraction \frac{-6}{-16} to lowest terms by extracting and canceling out 2.
t^{2}-\frac{7}{8}t+\left(-\frac{7}{16}\right)^{2}=\frac{3}{8}+\left(-\frac{7}{16}\right)^{2}
Divide -\frac{7}{8}, the coefficient of the x term, by 2 to get -\frac{7}{16}. Then add the square of -\frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-\frac{7}{8}t+\frac{49}{256}=\frac{3}{8}+\frac{49}{256}
Square -\frac{7}{16} by squaring both the numerator and the denominator of the fraction.
t^{2}-\frac{7}{8}t+\frac{49}{256}=\frac{145}{256}
Add \frac{3}{8} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t-\frac{7}{16}\right)^{2}=\frac{145}{256}
Factor t^{2}-\frac{7}{8}t+\frac{49}{256}. 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(t-\frac{7}{16}\right)^{2}}=\sqrt{\frac{145}{256}}
Take the square root of both sides of the equation.
t-\frac{7}{16}=\frac{\sqrt{145}}{16} t-\frac{7}{16}=-\frac{\sqrt{145}}{16}
Simplify.
t=\frac{\sqrt{145}+7}{16} t=\frac{7-\sqrt{145}}{16}
Add \frac{7}{16} to both sides of the equation.
x ^ 2 -\frac{7}{8}x -\frac{3}{8} = 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{7}{8} rs = -\frac{3}{8}
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{7}{16} - u s = \frac{7}{16} + u
Two numbers r and s sum up to \frac{7}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{8} = \frac{7}{16}. 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{7}{16} - u) (\frac{7}{16} + u) = -\frac{3}{8}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{8}
\frac{49}{256} - u^2 = -\frac{3}{8}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{3}{8}-\frac{49}{256} = -\frac{145}{256}
Simplify the expression by subtracting \frac{49}{256} on both sides
u^2 = \frac{145}{256} u = \pm\sqrt{\frac{145}{256}} = \pm \frac{\sqrt{145}}{16}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{7}{16} - \frac{\sqrt{145}}{16} = -0.315 s = \frac{7}{16} + \frac{\sqrt{145}}{16} = 1.190
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.