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1+x\times 8+x^{2}\times 15=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.
15x^{2}+8x+1=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=8 ab=15\times 1=15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 15x^{2}+ax+bx+1. To find a and b, set up a system to be solved.
1,15 3,5
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.
1+15=16 3+5=8
Calculate the sum for each pair.
a=3 b=5
The solution is the pair that gives sum 8.
\left(15x^{2}+3x\right)+\left(5x+1\right)
Rewrite 15x^{2}+8x+1 as \left(15x^{2}+3x\right)+\left(5x+1\right).
3x\left(5x+1\right)+5x+1
Factor out 3x in 15x^{2}+3x.
\left(5x+1\right)\left(3x+1\right)
Factor out common term 5x+1 by using distributive property.
x=-\frac{1}{5} x=-\frac{1}{3}
To find equation solutions, solve 5x+1=0 and 3x+1=0.
1+x\times 8+x^{2}\times 15=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.
15x^{2}+8x+1=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 15}}{2\times 15}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 15}}{2\times 15}
Square 8.
x=\frac{-8±\sqrt{64-60}}{2\times 15}
Multiply -4 times 15.
x=\frac{-8±\sqrt{4}}{2\times 15}
Add 64 to -60.
x=\frac{-8±2}{2\times 15}
Take the square root of 4.
x=\frac{-8±2}{30}
Multiply 2 times 15.
x=-\frac{6}{30}
Now solve the equation x=\frac{-8±2}{30} when ± is plus. Add -8 to 2.
x=-\frac{1}{5}
Reduce the fraction \frac{-6}{30} to lowest terms by extracting and canceling out 6.
x=-\frac{10}{30}
Now solve the equation x=\frac{-8±2}{30} when ± is minus. Subtract 2 from -8.
x=-\frac{1}{3}
Reduce the fraction \frac{-10}{30} to lowest terms by extracting and canceling out 10.
x=-\frac{1}{5} x=-\frac{1}{3}
The equation is now solved.
1+x\times 8+x^{2}\times 15=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.
x\times 8+x^{2}\times 15=-1
Subtract 1 from both sides. Anything subtracted from zero gives its negation.
15x^{2}+8x=-1
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{15x^{2}+8x}{15}=-\frac{1}{15}
Divide both sides by 15.
x^{2}+\frac{8}{15}x=-\frac{1}{15}
Dividing by 15 undoes the multiplication by 15.
x^{2}+\frac{8}{15}x+\left(\frac{4}{15}\right)^{2}=-\frac{1}{15}+\left(\frac{4}{15}\right)^{2}
Divide \frac{8}{15}, the coefficient of the x term, by 2 to get \frac{4}{15}. Then add the square of \frac{4}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{8}{15}x+\frac{16}{225}=-\frac{1}{15}+\frac{16}{225}
Square \frac{4}{15} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{8}{15}x+\frac{16}{225}=\frac{1}{225}
Add -\frac{1}{15} to \frac{16}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{4}{15}\right)^{2}=\frac{1}{225}
Factor x^{2}+\frac{8}{15}x+\frac{16}{225}. 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{4}{15}\right)^{2}}=\sqrt{\frac{1}{225}}
Take the square root of both sides of the equation.
x+\frac{4}{15}=\frac{1}{15} x+\frac{4}{15}=-\frac{1}{15}
Simplify.
x=-\frac{1}{5} x=-\frac{1}{3}
Subtract \frac{4}{15} from both sides of the equation.