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4t^{2}+7t+13=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{-7±\sqrt{7^{2}-4\times 4\times 13}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 7 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-7±\sqrt{49-4\times 4\times 13}}{2\times 4}
Square 7.
t=\frac{-7±\sqrt{49-16\times 13}}{2\times 4}
Multiply -4 times 4.
t=\frac{-7±\sqrt{49-208}}{2\times 4}
Multiply -16 times 13.
t=\frac{-7±\sqrt{-159}}{2\times 4}
Add 49 to -208.
t=\frac{-7±\sqrt{159}i}{2\times 4}
Take the square root of -159.
t=\frac{-7±\sqrt{159}i}{8}
Multiply 2 times 4.
t=\frac{-7+\sqrt{159}i}{8}
Now solve the equation t=\frac{-7±\sqrt{159}i}{8} when ± is plus. Add -7 to i\sqrt{159}.
t=\frac{-\sqrt{159}i-7}{8}
Now solve the equation t=\frac{-7±\sqrt{159}i}{8} when ± is minus. Subtract i\sqrt{159} from -7.
t=\frac{-7+\sqrt{159}i}{8} t=\frac{-\sqrt{159}i-7}{8}
The equation is now solved.
4t^{2}+7t+13=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.
4t^{2}+7t+13-13=-13
Subtract 13 from both sides of the equation.
4t^{2}+7t=-13
Subtracting 13 from itself leaves 0.
\frac{4t^{2}+7t}{4}=-\frac{13}{4}
Divide both sides by 4.
t^{2}+\frac{7}{4}t=-\frac{13}{4}
Dividing by 4 undoes the multiplication by 4.
t^{2}+\frac{7}{4}t+\left(\frac{7}{8}\right)^{2}=-\frac{13}{4}+\left(\frac{7}{8}\right)^{2}
Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}+\frac{7}{4}t+\frac{49}{64}=-\frac{13}{4}+\frac{49}{64}
Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.
t^{2}+\frac{7}{4}t+\frac{49}{64}=-\frac{159}{64}
Add -\frac{13}{4} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t+\frac{7}{8}\right)^{2}=-\frac{159}{64}
Factor t^{2}+\frac{7}{4}t+\frac{49}{64}. 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}{8}\right)^{2}}=\sqrt{-\frac{159}{64}}
Take the square root of both sides of the equation.
t+\frac{7}{8}=\frac{\sqrt{159}i}{8} t+\frac{7}{8}=-\frac{\sqrt{159}i}{8}
Simplify.
t=\frac{-7+\sqrt{159}i}{8} t=\frac{-\sqrt{159}i-7}{8}
Subtract \frac{7}{8} from both sides of the equation.
x ^ 2 +\frac{7}{4}x +\frac{13}{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{7}{4} rs = \frac{13}{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{7}{8} - u s = -\frac{7}{8} + u
Two numbers r and s sum up to -\frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{4} = -\frac{7}{8}. 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}{8} - u) (-\frac{7}{8} + u) = \frac{13}{4}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{13}{4}
\frac{49}{64} - u^2 = \frac{13}{4}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{13}{4}-\frac{49}{64} = \frac{159}{64}
Simplify the expression by subtracting \frac{49}{64} on both sides
u^2 = -\frac{159}{64} u = \pm\sqrt{-\frac{159}{64}} = \pm \frac{\sqrt{159}}{8}i
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}{8} - \frac{\sqrt{159}}{8}i = -0.875 - 1.576i s = -\frac{7}{8} + \frac{\sqrt{159}}{8}i = -0.875 + 1.576i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.