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7x^{2}+6x=4
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}+6x-4=4-4
Subtract 4 from both sides of the equation.
7x^{2}+6x-4=0
Subtracting 4 from itself leaves 0.
x=\frac{-6±\sqrt{6^{2}-4\times 7\left(-4\right)}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 6 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\times 7\left(-4\right)}}{2\times 7}
Square 6.
x=\frac{-6±\sqrt{36-28\left(-4\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-6±\sqrt{36+112}}{2\times 7}
Multiply -28 times -4.
x=\frac{-6±\sqrt{148}}{2\times 7}
Add 36 to 112.
x=\frac{-6±2\sqrt{37}}{2\times 7}
Take the square root of 148.
x=\frac{-6±2\sqrt{37}}{14}
Multiply 2 times 7.
x=\frac{2\sqrt{37}-6}{14}
Now solve the equation x=\frac{-6±2\sqrt{37}}{14} when ± is plus. Add -6 to 2\sqrt{37}.
x=\frac{\sqrt{37}-3}{7}
Divide -6+2\sqrt{37} by 14.
x=\frac{-2\sqrt{37}-6}{14}
Now solve the equation x=\frac{-6±2\sqrt{37}}{14} when ± is minus. Subtract 2\sqrt{37} from -6.
x=\frac{-\sqrt{37}-3}{7}
Divide -6-2\sqrt{37} by 14.
x=\frac{\sqrt{37}-3}{7} x=\frac{-\sqrt{37}-3}{7}
The equation is now solved.
7x^{2}+6x=4
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}+6x}{7}=\frac{4}{7}
Divide both sides by 7.
x^{2}+\frac{6}{7}x=\frac{4}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}+\frac{6}{7}x+\left(\frac{3}{7}\right)^{2}=\frac{4}{7}+\left(\frac{3}{7}\right)^{2}
Divide \frac{6}{7}, the coefficient of the x term, by 2 to get \frac{3}{7}. Then add the square of \frac{3}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{6}{7}x+\frac{9}{49}=\frac{4}{7}+\frac{9}{49}
Square \frac{3}{7} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{6}{7}x+\frac{9}{49}=\frac{37}{49}
Add \frac{4}{7} to \frac{9}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{3}{7}\right)^{2}=\frac{37}{49}
Factor x^{2}+\frac{6}{7}x+\frac{9}{49}. 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{3}{7}\right)^{2}}=\sqrt{\frac{37}{49}}
Take the square root of both sides of the equation.
x+\frac{3}{7}=\frac{\sqrt{37}}{7} x+\frac{3}{7}=-\frac{\sqrt{37}}{7}
Simplify.
x=\frac{\sqrt{37}-3}{7} x=\frac{-\sqrt{37}-3}{7}
Subtract \frac{3}{7} from both sides of the equation.