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5x^{2}+7x=3
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.
5x^{2}+7x-3=3-3
Subtract 3 from both sides of the equation.
5x^{2}+7x-3=0
Subtracting 3 from itself leaves 0.
x=\frac{-7±\sqrt{7^{2}-4\times 5\left(-3\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 7 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\times 5\left(-3\right)}}{2\times 5}
Square 7.
x=\frac{-7±\sqrt{49-20\left(-3\right)}}{2\times 5}
Multiply -4 times 5.
x=\frac{-7±\sqrt{49+60}}{2\times 5}
Multiply -20 times -3.
x=\frac{-7±\sqrt{109}}{2\times 5}
Add 49 to 60.
x=\frac{-7±\sqrt{109}}{10}
Multiply 2 times 5.
x=\frac{\sqrt{109}-7}{10}
Now solve the equation x=\frac{-7±\sqrt{109}}{10} when ± is plus. Add -7 to \sqrt{109}.
x=\frac{-\sqrt{109}-7}{10}
Now solve the equation x=\frac{-7±\sqrt{109}}{10} when ± is minus. Subtract \sqrt{109} from -7.
x=\frac{\sqrt{109}-7}{10} x=\frac{-\sqrt{109}-7}{10}
The equation is now solved.
5x^{2}+7x=3
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{5x^{2}+7x}{5}=\frac{3}{5}
Divide both sides by 5.
x^{2}+\frac{7}{5}x=\frac{3}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+\frac{7}{5}x+\left(\frac{7}{10}\right)^{2}=\frac{3}{5}+\left(\frac{7}{10}\right)^{2}
Divide \frac{7}{5}, the coefficient of the x term, by 2 to get \frac{7}{10}. Then add the square of \frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{3}{5}+\frac{49}{100}
Square \frac{7}{10} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{109}{100}
Add \frac{3}{5} to \frac{49}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{7}{10}\right)^{2}=\frac{109}{100}
Factor x^{2}+\frac{7}{5}x+\frac{49}{100}. 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{7}{10}\right)^{2}}=\sqrt{\frac{109}{100}}
Take the square root of both sides of the equation.
x+\frac{7}{10}=\frac{\sqrt{109}}{10} x+\frac{7}{10}=-\frac{\sqrt{109}}{10}
Simplify.
x=\frac{\sqrt{109}-7}{10} x=\frac{-\sqrt{109}-7}{10}
Subtract \frac{7}{10} from both sides of the equation.