Solve for q
q = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333
q=-\frac{3}{7}\approx -0.428571429
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Polynomial5 problems similar to: 21 q ^ { 2 } + 37 q + 12 = 0Similar Problems from Web Search
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a+b=37 ab=21\times 12=252
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 21q^{2}+aq+bq+12. To find a and b, set up a system to be solved.
1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18
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 252.
1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32
Calculate the sum for each pair.
a=9 b=28
The solution is the pair that gives sum 37.
\left(21q^{2}+9q\right)+\left(28q+12\right)
Rewrite 21q^{2}+37q+12 as \left(21q^{2}+9q\right)+\left(28q+12\right).
3q\left(7q+3\right)+4\left(7q+3\right)
Factor out 3q in the first and 4 in the second group.
\left(7q+3\right)\left(3q+4\right)
Factor out common term 7q+3 by using distributive property.
q=-\frac{3}{7} q=-\frac{4}{3}
To find equation solutions, solve 7q+3=0 and 3q+4=0.
21q^{2}+37q+12=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.
q=\frac{-37±\sqrt{37^{2}-4\times 21\times 12}}{2\times 21}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, 37 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
q=\frac{-37±\sqrt{1369-4\times 21\times 12}}{2\times 21}
Square 37.
q=\frac{-37±\sqrt{1369-84\times 12}}{2\times 21}
Multiply -4 times 21.
q=\frac{-37±\sqrt{1369-1008}}{2\times 21}
Multiply -84 times 12.
q=\frac{-37±\sqrt{361}}{2\times 21}
Add 1369 to -1008.
q=\frac{-37±19}{2\times 21}
Take the square root of 361.
q=\frac{-37±19}{42}
Multiply 2 times 21.
q=-\frac{18}{42}
Now solve the equation q=\frac{-37±19}{42} when ± is plus. Add -37 to 19.
q=-\frac{3}{7}
Reduce the fraction \frac{-18}{42} to lowest terms by extracting and canceling out 6.
q=-\frac{56}{42}
Now solve the equation q=\frac{-37±19}{42} when ± is minus. Subtract 19 from -37.
q=-\frac{4}{3}
Reduce the fraction \frac{-56}{42} to lowest terms by extracting and canceling out 14.
q=-\frac{3}{7} q=-\frac{4}{3}
The equation is now solved.
21q^{2}+37q+12=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.
21q^{2}+37q+12-12=-12
Subtract 12 from both sides of the equation.
21q^{2}+37q=-12
Subtracting 12 from itself leaves 0.
\frac{21q^{2}+37q}{21}=-\frac{12}{21}
Divide both sides by 21.
q^{2}+\frac{37}{21}q=-\frac{12}{21}
Dividing by 21 undoes the multiplication by 21.
q^{2}+\frac{37}{21}q=-\frac{4}{7}
Reduce the fraction \frac{-12}{21} to lowest terms by extracting and canceling out 3.
q^{2}+\frac{37}{21}q+\left(\frac{37}{42}\right)^{2}=-\frac{4}{7}+\left(\frac{37}{42}\right)^{2}
Divide \frac{37}{21}, the coefficient of the x term, by 2 to get \frac{37}{42}. Then add the square of \frac{37}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
q^{2}+\frac{37}{21}q+\frac{1369}{1764}=-\frac{4}{7}+\frac{1369}{1764}
Square \frac{37}{42} by squaring both the numerator and the denominator of the fraction.
q^{2}+\frac{37}{21}q+\frac{1369}{1764}=\frac{361}{1764}
Add -\frac{4}{7} to \frac{1369}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(q+\frac{37}{42}\right)^{2}=\frac{361}{1764}
Factor q^{2}+\frac{37}{21}q+\frac{1369}{1764}. 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(q+\frac{37}{42}\right)^{2}}=\sqrt{\frac{361}{1764}}
Take the square root of both sides of the equation.
q+\frac{37}{42}=\frac{19}{42} q+\frac{37}{42}=-\frac{19}{42}
Simplify.
q=-\frac{3}{7} q=-\frac{4}{3}
Subtract \frac{37}{42} from both sides of the equation.
x ^ 2 +\frac{37}{21}x +\frac{4}{7} = 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{37}{21} rs = \frac{4}{7}
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{37}{42} - u s = -\frac{37}{42} + u
Two numbers r and s sum up to -\frac{37}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{37}{21} = -\frac{37}{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{37}{42} - u) (-\frac{37}{42} + u) = \frac{4}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{7}
\frac{1369}{1764} - u^2 = \frac{4}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{4}{7}-\frac{1369}{1764} = -\frac{361}{1764}
Simplify the expression by subtracting \frac{1369}{1764} on both sides
u^2 = \frac{361}{1764} u = \pm\sqrt{\frac{361}{1764}} = \pm \frac{19}{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{37}{42} - \frac{19}{42} = -1.333 s = -\frac{37}{42} + \frac{19}{42} = -0.429
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
