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±\frac{15}{8},±\frac{15}{4},±\frac{45}{8},±\frac{15}{2},±\frac{45}{4},±15,±\frac{45}{2},±45,±\frac{5}{8},±\frac{5}{4},±\frac{5}{2},±5,±\frac{3}{8},±\frac{3}{4},±\frac{9}{8},±\frac{3}{2},±\frac{9}{4},±3,±\frac{9}{2},±9,±\frac{5}{24},±\frac{5}{12},±\frac{5}{6},±\frac{5}{3},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1,±\frac{1}{24},±\frac{1}{12},±\frac{1}{6},±\frac{1}{3}
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 45 and q divides the leading coefficient 24. List all candidates \frac{p}{q}.
x=\frac{3}{4}
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
6x^{2}+x-15=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 24x^{3}-14x^{2}-63x+45 by 4\left(x-\frac{3}{4}\right)=4x-3 to get 6x^{2}+x-15. Solve the equation where the result equals to 0.
x=\frac{-1±\sqrt{1^{2}-4\times 6\left(-15\right)}}{2\times 6}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 6 for a, 1 for b, and -15 for c in the quadratic formula.
x=\frac{-1±19}{12}
Do the calculations.
x=-\frac{5}{3} x=\frac{3}{2}
Solve the equation 6x^{2}+x-15=0 when ± is plus and when ± is minus.
x=\frac{3}{4} x=-\frac{5}{3} x=\frac{3}{2}
List all found solutions.