Factor
\left(7x-4\right)\left(x+2\right)
Evaluate
\left(7x-4\right)\left(x+2\right)
Graph
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a+b=10 ab=7\left(-8\right)=-56
Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-8. To find a and b, set up a system to be solved.
-1,56 -2,28 -4,14 -7,8
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 -56.
-1+56=55 -2+28=26 -4+14=10 -7+8=1
Calculate the sum for each pair.
a=-4 b=14
The solution is the pair that gives sum 10.
\left(7x^{2}-4x\right)+\left(14x-8\right)
Rewrite 7x^{2}+10x-8 as \left(7x^{2}-4x\right)+\left(14x-8\right).
x\left(7x-4\right)+2\left(7x-4\right)
Factor out x in the first and 2 in the second group.
\left(7x-4\right)\left(x+2\right)
Factor out common term 7x-4 by using distributive property.
7x^{2}+10x-8=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{-10±\sqrt{10^{2}-4\times 7\left(-8\right)}}{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.
x=\frac{-10±\sqrt{100-4\times 7\left(-8\right)}}{2\times 7}
Square 10.
x=\frac{-10±\sqrt{100-28\left(-8\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-10±\sqrt{100+224}}{2\times 7}
Multiply -28 times -8.
x=\frac{-10±\sqrt{324}}{2\times 7}
Add 100 to 224.
x=\frac{-10±18}{2\times 7}
Take the square root of 324.
x=\frac{-10±18}{14}
Multiply 2 times 7.
x=\frac{8}{14}
Now solve the equation x=\frac{-10±18}{14} when ± is plus. Add -10 to 18.
x=\frac{4}{7}
Reduce the fraction \frac{8}{14} to lowest terms by extracting and canceling out 2.
x=-\frac{28}{14}
Now solve the equation x=\frac{-10±18}{14} when ± is minus. Subtract 18 from -10.
x=-2
Divide -28 by 14.
7x^{2}+10x-8=7\left(x-\frac{4}{7}\right)\left(x-\left(-2\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{7} for x_{1} and -2 for x_{2}.
7x^{2}+10x-8=7\left(x-\frac{4}{7}\right)\left(x+2\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
7x^{2}+10x-8=7\times \frac{7x-4}{7}\left(x+2\right)
Subtract \frac{4}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
7x^{2}+10x-8=\left(7x-4\right)\left(x+2\right)
Cancel out 7, the greatest common factor in 7 and 7.
x ^ 2 +\frac{10}{7}x -\frac{8}{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{10}{7} rs = -\frac{8}{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{5}{7} - u s = -\frac{5}{7} + u
Two numbers r and s sum up to -\frac{10}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{10}{7} = -\frac{5}{7}. 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{5}{7} - u) (-\frac{5}{7} + u) = -\frac{8}{7}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{7}
\frac{25}{49} - u^2 = -\frac{8}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{8}{7}-\frac{25}{49} = -\frac{81}{49}
Simplify the expression by subtracting \frac{25}{49} on both sides
u^2 = \frac{81}{49} u = \pm\sqrt{\frac{81}{49}} = \pm \frac{9}{7}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{5}{7} - \frac{9}{7} = -2 s = -\frac{5}{7} + \frac{9}{7} = 0.571
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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