Solve for x
x=1
x=\frac{6}{7}\approx 0.857142857
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a+b=-13 ab=7\times 6=42
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx+6. To find a and b, set up a system to be solved.
-1,-42 -2,-21 -3,-14 -6,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.
-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13
Calculate the sum for each pair.
a=-7 b=-6
The solution is the pair that gives sum -13.
\left(7x^{2}-7x\right)+\left(-6x+6\right)
Rewrite 7x^{2}-13x+6 as \left(7x^{2}-7x\right)+\left(-6x+6\right).
7x\left(x-1\right)-6\left(x-1\right)
Factor out 7x in the first and -6 in the second group.
\left(x-1\right)\left(7x-6\right)
Factor out common term x-1 by using distributive property.
x=1 x=\frac{6}{7}
To find equation solutions, solve x-1=0 and 7x-6=0.
7x^{2}-13x+6=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(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 7\times 6}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -13 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-13\right)±\sqrt{169-4\times 7\times 6}}{2\times 7}
Square -13.
x=\frac{-\left(-13\right)±\sqrt{169-28\times 6}}{2\times 7}
Multiply -4 times 7.
x=\frac{-\left(-13\right)±\sqrt{169-168}}{2\times 7}
Multiply -28 times 6.
x=\frac{-\left(-13\right)±\sqrt{1}}{2\times 7}
Add 169 to -168.
x=\frac{-\left(-13\right)±1}{2\times 7}
Take the square root of 1.
x=\frac{13±1}{2\times 7}
The opposite of -13 is 13.
x=\frac{13±1}{14}
Multiply 2 times 7.
x=\frac{14}{14}
Now solve the equation x=\frac{13±1}{14} when ± is plus. Add 13 to 1.
x=1
Divide 14 by 14.
x=\frac{12}{14}
Now solve the equation x=\frac{13±1}{14} when ± is minus. Subtract 1 from 13.
x=\frac{6}{7}
Reduce the fraction \frac{12}{14} to lowest terms by extracting and canceling out 2.
x=1 x=\frac{6}{7}
The equation is now solved.
7x^{2}-13x+6=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.
7x^{2}-13x+6-6=-6
Subtract 6 from both sides of the equation.
7x^{2}-13x=-6
Subtracting 6 from itself leaves 0.
\frac{7x^{2}-13x}{7}=-\frac{6}{7}
Divide both sides by 7.
x^{2}-\frac{13}{7}x=-\frac{6}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}-\frac{13}{7}x+\left(-\frac{13}{14}\right)^{2}=-\frac{6}{7}+\left(-\frac{13}{14}\right)^{2}
Divide -\frac{13}{7}, the coefficient of the x term, by 2 to get -\frac{13}{14}. Then add the square of -\frac{13}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{13}{7}x+\frac{169}{196}=-\frac{6}{7}+\frac{169}{196}
Square -\frac{13}{14} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{13}{7}x+\frac{169}{196}=\frac{1}{196}
Add -\frac{6}{7} to \frac{169}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{13}{14}\right)^{2}=\frac{1}{196}
Factor x^{2}-\frac{13}{7}x+\frac{169}{196}. 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{13}{14}\right)^{2}}=\sqrt{\frac{1}{196}}
Take the square root of both sides of the equation.
x-\frac{13}{14}=\frac{1}{14} x-\frac{13}{14}=-\frac{1}{14}
Simplify.
x=1 x=\frac{6}{7}
Add \frac{13}{14} to both sides of the equation.
x ^ 2 -\frac{13}{7}x +\frac{6}{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 7
r + s = \frac{13}{7} rs = \frac{6}{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{13}{14} - u s = \frac{13}{14} + u
Two numbers r and s sum up to \frac{13}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{7} = \frac{13}{14}. 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}{14} - u) (\frac{13}{14} + u) = \frac{6}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{6}{7}
\frac{169}{196} - u^2 = \frac{6}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{6}{7}-\frac{169}{196} = -\frac{1}{196}
Simplify the expression by subtracting \frac{169}{196} on both sides
u^2 = \frac{1}{196} u = \pm\sqrt{\frac{1}{196}} = \pm \frac{1}{14}
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}{14} - \frac{1}{14} = 0.857 s = \frac{13}{14} + \frac{1}{14} = 1.000
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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