Skip to main content
Factor
Tick mark Image
Evaluate
Tick mark Image

Similar Problems from Web Search

Share

7m^{2}-25m+6=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.
m=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 7\times 6}}{2\times 7}
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.
m=\frac{-\left(-25\right)±\sqrt{625-4\times 7\times 6}}{2\times 7}
Square -25.
m=\frac{-\left(-25\right)±\sqrt{625-28\times 6}}{2\times 7}
Multiply -4 times 7.
m=\frac{-\left(-25\right)±\sqrt{625-168}}{2\times 7}
Multiply -28 times 6.
m=\frac{-\left(-25\right)±\sqrt{457}}{2\times 7}
Add 625 to -168.
m=\frac{25±\sqrt{457}}{2\times 7}
The opposite of -25 is 25.
m=\frac{25±\sqrt{457}}{14}
Multiply 2 times 7.
m=\frac{\sqrt{457}+25}{14}
Now solve the equation m=\frac{25±\sqrt{457}}{14} when ± is plus. Add 25 to \sqrt{457}.
m=\frac{25-\sqrt{457}}{14}
Now solve the equation m=\frac{25±\sqrt{457}}{14} when ± is minus. Subtract \sqrt{457} from 25.
7m^{2}-25m+6=7\left(m-\frac{\sqrt{457}+25}{14}\right)\left(m-\frac{25-\sqrt{457}}{14}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{25+\sqrt{457}}{14} for x_{1} and \frac{25-\sqrt{457}}{14} for x_{2}.
x ^ 2 -\frac{25}{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{25}{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{25}{14} - u s = \frac{25}{14} + u
Two numbers r and s sum up to \frac{25}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{25}{7} = \frac{25}{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{25}{14} - u) (\frac{25}{14} + u) = \frac{6}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{6}{7}
\frac{625}{196} - u^2 = \frac{6}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{6}{7}-\frac{625}{196} = -\frac{457}{196}
Simplify the expression by subtracting \frac{625}{196} on both sides
u^2 = \frac{457}{196} u = \pm\sqrt{\frac{457}{196}} = \pm \frac{\sqrt{457}}{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{25}{14} - \frac{\sqrt{457}}{14} = 0.259 s = \frac{25}{14} + \frac{\sqrt{457}}{14} = 3.313
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.