What Is the Positive Solution to 0 1 3x 2-3
a+b=13 ab=21\left(-20\right)=-420
Factor the expression by grouping. First, the expression needs to be rewritten as 21x^{2}+ax+bx-20. To find a and b, set up a system to be solved.
-1,420 -2,210 -3,140 -4,105 -5,84 -6,70 -7,60 -10,42 -12,35 -14,30 -15,28 -20,21
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -420.
-1+420=419 -2+210=208 -3+140=137 -4+105=101 -5+84=79 -6+70=64 -7+60=53 -10+42=32 -12+35=23 -14+30=16 -15+28=13 -20+21=1
Calculate the sum for each pair.
a=-15 b=28
The solution is the pair that gives sum 13.
\left(21x^{2}-15x\right)+\left(28x-20\right)
Rewrite 21x^{2}+13x-20 as \left(21x^{2}-15x\right)+\left(28x-20\right).
3x\left(7x-5\right)+4\left(7x-5\right)
Factor out 3x in the first and 4 in the second group.
\left(7x-5\right)\left(3x+4\right)
Factor out common term 7x-5 by using distributive property.
21x^{2}+13x-20=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-13±\sqrt{13^{2}-4\times 21\left(-20\right)}}{2\times 21}
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{-13±\sqrt{169-4\times 21\left(-20\right)}}{2\times 21}
Square 13.
x=\frac{-13±\sqrt{169-84\left(-20\right)}}{2\times 21}
Multiply -4 times 21.
x=\frac{-13±\sqrt{169+1680}}{2\times 21}
Multiply -84 times -20.
x=\frac{-13±\sqrt{1849}}{2\times 21}
Add 169 to 1680.
x=\frac{-13±43}{2\times 21}
Take the square root of 1849.
x=\frac{-13±43}{42}
Multiply 2 times 21.
x=\frac{30}{42}
Now solve the equation x=\frac{-13±43}{42} when ± is plus. Add -13 to 43.
x=\frac{5}{7}
Reduce the fraction \frac{30}{42} to lowest terms by extracting and canceling out 6.
x=\frac{-56}{42}
Now solve the equation x=\frac{-13±43}{42} when ± is minus. Subtract 43 from -13.
x=-\frac{4}{3}
Reduce the fraction \frac{-56}{42} to lowest terms by extracting and canceling out 14.
21x^{2}+13x-20=21\left(x-\frac{5}{7}\right)\left(x-\left(-\frac{4}{3}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{7} for x_{1} and -\frac{4}{3} for x_{2}.
21x^{2}+13x-20=21\left(x-\frac{5}{7}\right)\left(x+\frac{4}{3}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
21x^{2}+13x-20=21\times \left(\frac{7x-5}{7}\right)\left(x+\frac{4}{3}\right)
Subtract \frac{5}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
21x^{2}+13x-20=21\times \left(\frac{7x-5}{7}\right)\times \left(\frac{3x+4}{3}\right)
Add \frac{4}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
21x^{2}+13x-20=21\times \left(\frac{\left(7x-5\right)\left(3x+4\right)}{7\times 3}\right)
Multiply \frac{7x-5}{7} times \frac{3x+4}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
21x^{2}+13x-20=21\times \left(\frac{\left(7x-5\right)\left(3x+4\right)}{21}\right)
Multiply 7 times 3.
21x^{2}+13x-20=\left(7x-5\right)\left(3x+4\right)
Cancel out 21, the greatest common factor in 21 and 21.
x ^ 2 +\frac{13}{21}x -\frac{20}{21} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 21
r + s = -\frac{13}{21} rs = -\frac{20}{21}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{13}{42} - u s = -\frac{13}{42} + u
Two numbers r and s sum up to -\frac{13}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{21} = -\frac{13}{42}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{13}{42} - u) (-\frac{13}{42} + u) = -\frac{20}{21}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{20}{21}
\frac{169}{1764} - u^2 = -\frac{20}{21}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{20}{21}-\frac{169}{1764} = -\frac{1849}{1764}
Simplify the expression by subtracting \frac{169}{1764} on both sides
u^2 = \frac{1849}{1764} u = \pm\sqrt{\frac{1849}{1764}} = \pm \frac{43}{42}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{13}{42} - \frac{43}{42} = -1.333 s = -\frac{13}{42} + \frac{43}{42} = 0.714
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
What Is the Positive Solution to 0 1 3x 2-3
Source: https://mathsolver.microsoft.com/en/solve-problem/21%20x%20%5E%20%7B%202%20%7D%20%2B%2013%20x%20-%2020