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1.5x^{2}+10x-37.5=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{-10±\sqrt{10^{2}-4\times 1.5\left(-37.5\right)}}{2\times 1.5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5 for a, 10 for b, and -37.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\times 1.5\left(-37.5\right)}}{2\times 1.5}
Square 10.
x=\frac{-10±\sqrt{100-6\left(-37.5\right)}}{2\times 1.5}
Multiply -4 times 1.5.
x=\frac{-10±\sqrt{100+225}}{2\times 1.5}
Multiply -6 times -37.5.
x=\frac{-10±\sqrt{325}}{2\times 1.5}
Add 100 to 225.
x=\frac{-10±5\sqrt{13}}{2\times 1.5}
Take the square root of 325.
x=\frac{-10±5\sqrt{13}}{3}
Multiply 2 times 1.5.
x=\frac{5\sqrt{13}-10}{3}
Now solve the equation x=\frac{-10±5\sqrt{13}}{3} when ± is plus. Add -10 to 5\sqrt{13}.
x=\frac{-5\sqrt{13}-10}{3}
Now solve the equation x=\frac{-10±5\sqrt{13}}{3} when ± is minus. Subtract 5\sqrt{13} from -10.
x=\frac{5\sqrt{13}-10}{3} x=\frac{-5\sqrt{13}-10}{3}
The equation is now solved.
1.5x^{2}+10x-37.5=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.
1.5x^{2}+10x-37.5-\left(-37.5\right)=-\left(-37.5\right)
Add 37.5 to both sides of the equation.
1.5x^{2}+10x=-\left(-37.5\right)
Subtracting -37.5 from itself leaves 0.
1.5x^{2}+10x=37.5
Subtract -37.5 from 0.
\frac{1.5x^{2}+10x}{1.5}=\frac{37.5}{1.5}
Divide both sides of the equation by 1.5, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\frac{10}{1.5}x=\frac{37.5}{1.5}
Dividing by 1.5 undoes the multiplication by 1.5.
x^{2}+\frac{20}{3}x=\frac{37.5}{1.5}
Divide 10 by 1.5 by multiplying 10 by the reciprocal of 1.5.
x^{2}+\frac{20}{3}x=25
Divide 37.5 by 1.5 by multiplying 37.5 by the reciprocal of 1.5.
x^{2}+\frac{20}{3}x+\frac{10}{3}^{2}=25+\frac{10}{3}^{2}
Divide \frac{20}{3}, the coefficient of the x term, by 2 to get \frac{10}{3}. Then add the square of \frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{20}{3}x+\frac{100}{9}=25+\frac{100}{9}
Square \frac{10}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{20}{3}x+\frac{100}{9}=\frac{325}{9}
Add 25 to \frac{100}{9}.
\left(x+\frac{10}{3}\right)^{2}=\frac{325}{9}
Factor x^{2}+\frac{20}{3}x+\frac{100}{9}. 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{10}{3}\right)^{2}}=\sqrt{\frac{325}{9}}
Take the square root of both sides of the equation.
x+\frac{10}{3}=\frac{5\sqrt{13}}{3} x+\frac{10}{3}=-\frac{5\sqrt{13}}{3}
Simplify.
x=\frac{5\sqrt{13}-10}{3} x=\frac{-5\sqrt{13}-10}{3}
Subtract \frac{10}{3} from both sides of the equation.