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8x^{2}+8x-75=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.
x=\frac{-8±\sqrt{8^{2}-4\times 8\left(-75\right)}}{2\times 8}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 8 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 8\left(-75\right)}}{2\times 8}
Square 8.
x=\frac{-8±\sqrt{64-32\left(-75\right)}}{2\times 8}
Multiply -4 times 8.
x=\frac{-8±\sqrt{64+2400}}{2\times 8}
Multiply -32 times -75.
x=\frac{-8±\sqrt{2464}}{2\times 8}
Add 64 to 2400.
x=\frac{-8±4\sqrt{154}}{2\times 8}
Take the square root of 2464.
x=\frac{-8±4\sqrt{154}}{16}
Multiply 2 times 8.
x=\frac{4\sqrt{154}-8}{16}
Now solve the equation x=\frac{-8±4\sqrt{154}}{16} when ± is plus. Add -8 to 4\sqrt{154}.
x=\frac{\sqrt{154}}{4}-\frac{1}{2}
Divide -8+4\sqrt{154} by 16.
x=\frac{-4\sqrt{154}-8}{16}
Now solve the equation x=\frac{-8±4\sqrt{154}}{16} when ± is minus. Subtract 4\sqrt{154} from -8.
x=-\frac{\sqrt{154}}{4}-\frac{1}{2}
Divide -8-4\sqrt{154} by 16.
x=\frac{\sqrt{154}}{4}-\frac{1}{2} x=-\frac{\sqrt{154}}{4}-\frac{1}{2}
The equation is now solved.
8x^{2}+8x-75=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.
8x^{2}+8x-75-\left(-75\right)=-\left(-75\right)
Add 75 to both sides of the equation.
8x^{2}+8x=-\left(-75\right)
Subtracting -75 from itself leaves 0.
8x^{2}+8x=75
Subtract -75 from 0.
\frac{8x^{2}+8x}{8}=\frac{75}{8}
Divide both sides by 8.
x^{2}+\frac{8}{8}x=\frac{75}{8}
Dividing by 8 undoes the multiplication by 8.
x^{2}+x=\frac{75}{8}
Divide 8 by 8.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{75}{8}+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+x+\frac{1}{4}=\frac{75}{8}+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+x+\frac{1}{4}=\frac{77}{8}
Add \frac{75}{8} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{2}\right)^{2}=\frac{77}{8}
Factor x^{2}+x+\frac{1}{4}. 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{1}{2}\right)^{2}}=\sqrt{\frac{77}{8}}
Take the square root of both sides of the equation.
x+\frac{1}{2}=\frac{\sqrt{154}}{4} x+\frac{1}{2}=-\frac{\sqrt{154}}{4}
Simplify.
x=\frac{\sqrt{154}}{4}-\frac{1}{2} x=-\frac{\sqrt{154}}{4}-\frac{1}{2}
Subtract \frac{1}{2} from both sides of the equation.