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7x^{2}+13x=5
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.
7x^{2}+13x-5=5-5
Subtract 5 from both sides of the equation.
7x^{2}+13x-5=0
Subtracting 5 from itself leaves 0.
x=\frac{-13±\sqrt{13^{2}-4\times 7\left(-5\right)}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 13 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-13±\sqrt{169-4\times 7\left(-5\right)}}{2\times 7}
Square 13.
x=\frac{-13±\sqrt{169-28\left(-5\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-13±\sqrt{169+140}}{2\times 7}
Multiply -28 times -5.
x=\frac{-13±\sqrt{309}}{2\times 7}
Add 169 to 140.
x=\frac{-13±\sqrt{309}}{14}
Multiply 2 times 7.
x=\frac{\sqrt{309}-13}{14}
Now solve the equation x=\frac{-13±\sqrt{309}}{14} when ± is plus. Add -13 to \sqrt{309}.
x=\frac{-\sqrt{309}-13}{14}
Now solve the equation x=\frac{-13±\sqrt{309}}{14} when ± is minus. Subtract \sqrt{309} from -13.
x=\frac{\sqrt{309}-13}{14} x=\frac{-\sqrt{309}-13}{14}
The equation is now solved.
7x^{2}+13x=5
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.
\frac{7x^{2}+13x}{7}=\frac{5}{7}
Divide both sides by 7.
x^{2}+\frac{13}{7}x=\frac{5}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}+\frac{13}{7}x+\left(\frac{13}{14}\right)^{2}=\frac{5}{7}+\left(\frac{13}{14}\right)^{2}
Divide \frac{13}{7}, the coefficient of the x term, by 2 to get \frac{13}{14}. Then add the square of \frac{13}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{13}{7}x+\frac{169}{196}=\frac{5}{7}+\frac{169}{196}
Square \frac{13}{14} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{13}{7}x+\frac{169}{196}=\frac{309}{196}
Add \frac{5}{7} to \frac{169}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{13}{14}\right)^{2}=\frac{309}{196}
Factor x^{2}+\frac{13}{7}x+\frac{169}{196}. 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{13}{14}\right)^{2}}=\sqrt{\frac{309}{196}}
Take the square root of both sides of the equation.
x+\frac{13}{14}=\frac{\sqrt{309}}{14} x+\frac{13}{14}=-\frac{\sqrt{309}}{14}
Simplify.
x=\frac{\sqrt{309}-13}{14} x=\frac{-\sqrt{309}-13}{14}
Subtract \frac{13}{14} from both sides of the equation.