Skip to main content
Solve for x
Tick mark Image
Graph

Similar Problems from Web Search

Share

21x^{2}-8x-4=0
Divide both sides by 3.
a+b=-8 ab=21\left(-4\right)=-84
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 21x^{2}+ax+bx-4. To find a and b, set up a system to be solved.
1,-84 2,-42 3,-28 4,-21 6,-14 7,-12
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -84.
1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5
Calculate the sum for each pair.
a=-14 b=6
The solution is the pair that gives sum -8.
\left(21x^{2}-14x\right)+\left(6x-4\right)
Rewrite 21x^{2}-8x-4 as \left(21x^{2}-14x\right)+\left(6x-4\right).
7x\left(3x-2\right)+2\left(3x-2\right)
Factor out 7x in the first and 2 in the second group.
\left(3x-2\right)\left(7x+2\right)
Factor out common term 3x-2 by using distributive property.
x=\frac{2}{3} x=-\frac{2}{7}
To find equation solutions, solve 3x-2=0 and 7x+2=0.
63x^{2}-24x-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.
x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 63\left(-12\right)}}{2\times 63}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 63 for a, -24 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-24\right)±\sqrt{576-4\times 63\left(-12\right)}}{2\times 63}
Square -24.
x=\frac{-\left(-24\right)±\sqrt{576-252\left(-12\right)}}{2\times 63}
Multiply -4 times 63.
x=\frac{-\left(-24\right)±\sqrt{576+3024}}{2\times 63}
Multiply -252 times -12.
x=\frac{-\left(-24\right)±\sqrt{3600}}{2\times 63}
Add 576 to 3024.
x=\frac{-\left(-24\right)±60}{2\times 63}
Take the square root of 3600.
x=\frac{24±60}{2\times 63}
The opposite of -24 is 24.
x=\frac{24±60}{126}
Multiply 2 times 63.
x=\frac{84}{126}
Now solve the equation x=\frac{24±60}{126} when ± is plus. Add 24 to 60.
x=\frac{2}{3}
Reduce the fraction \frac{84}{126} to lowest terms by extracting and canceling out 42.
x=-\frac{36}{126}
Now solve the equation x=\frac{24±60}{126} when ± is minus. Subtract 60 from 24.
x=-\frac{2}{7}
Reduce the fraction \frac{-36}{126} to lowest terms by extracting and canceling out 18.
x=\frac{2}{3} x=-\frac{2}{7}
The equation is now solved.
63x^{2}-24x-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.
63x^{2}-24x-12-\left(-12\right)=-\left(-12\right)
Add 12 to both sides of the equation.
63x^{2}-24x=-\left(-12\right)
Subtracting -12 from itself leaves 0.
63x^{2}-24x=12
Subtract -12 from 0.
\frac{63x^{2}-24x}{63}=\frac{12}{63}
Divide both sides by 63.
x^{2}+\left(-\frac{24}{63}\right)x=\frac{12}{63}
Dividing by 63 undoes the multiplication by 63.
x^{2}-\frac{8}{21}x=\frac{12}{63}
Reduce the fraction \frac{-24}{63} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{8}{21}x=\frac{4}{21}
Reduce the fraction \frac{12}{63} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{8}{21}x+\left(-\frac{4}{21}\right)^{2}=\frac{4}{21}+\left(-\frac{4}{21}\right)^{2}
Divide -\frac{8}{21}, the coefficient of the x term, by 2 to get -\frac{4}{21}. Then add the square of -\frac{4}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{8}{21}x+\frac{16}{441}=\frac{4}{21}+\frac{16}{441}
Square -\frac{4}{21} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{8}{21}x+\frac{16}{441}=\frac{100}{441}
Add \frac{4}{21} to \frac{16}{441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{4}{21}\right)^{2}=\frac{100}{441}
Factor x^{2}-\frac{8}{21}x+\frac{16}{441}. 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}{21}\right)^{2}}=\sqrt{\frac{100}{441}}
Take the square root of both sides of the equation.
x-\frac{4}{21}=\frac{10}{21} x-\frac{4}{21}=-\frac{10}{21}
Simplify.
x=\frac{2}{3} x=-\frac{2}{7}
Add \frac{4}{21} to both sides of the equation.
x ^ 2 -\frac{8}{21}x -\frac{4}{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 63
r + s = \frac{8}{21} rs = -\frac{4}{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{4}{21} - u s = \frac{4}{21} + u
Two numbers r and s sum up to \frac{8}{21} exactly when the average of the two numbers is \frac{1}{2}*\frac{8}{21} = \frac{4}{21}. 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{4}{21} - u) (\frac{4}{21} + u) = -\frac{4}{21}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{21}
\frac{16}{441} - u^2 = -\frac{4}{21}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{21}-\frac{16}{441} = -\frac{100}{441}
Simplify the expression by subtracting \frac{16}{441} on both sides
u^2 = \frac{100}{441} u = \pm\sqrt{\frac{100}{441}} = \pm \frac{10}{21}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{4}{21} - \frac{10}{21} = -0.286 s = \frac{4}{21} + \frac{10}{21} = 0.667
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.