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a+b=-7 ab=4\left(-2\right)=-8
Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-2. To find a and b, set up a system to be solved.
1,-8 2,-4
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 -8.
1-8=-7 2-4=-2
Calculate the sum for each pair.
a=-8 b=1
The solution is the pair that gives sum -7.
\left(4x^{2}-8x\right)+\left(x-2\right)
Rewrite 4x^{2}-7x-2 as \left(4x^{2}-8x\right)+\left(x-2\right).
4x\left(x-2\right)+x-2
Factor out 4x in 4x^{2}-8x.
\left(x-2\right)\left(4x+1\right)
Factor out common term x-2 by using distributive property.
4x^{2}-7x-2=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{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-2\right)}}{2\times 4}
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(-7\right)±\sqrt{49-4\times 4\left(-2\right)}}{2\times 4}
Square -7.
x=\frac{-\left(-7\right)±\sqrt{49-16\left(-2\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-7\right)±\sqrt{49+32}}{2\times 4}
Multiply -16 times -2.
x=\frac{-\left(-7\right)±\sqrt{81}}{2\times 4}
x=\frac{-\left(-7\right)±9}{2\times 4}
Take the square root of 81.
x=\frac{7±9}{2\times 4}
The opposite of -7 is 7.
x=\frac{7±9}{8}
Multiply 2 times 4.
x=\frac{16}{8}
Now solve the equation x=\frac{7±9}{8} when ± is plus. Add 7 to 9.
x=2
Divide 16 by 8.
x=\frac{-2}{8}
Now solve the equation x=\frac{7±9}{8} when ± is minus. Subtract 9 from 7.
x=-\frac{1}{4}
Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.
4x^{2}-7x-2=4\left(x-2\right)\left(x-\left(-\frac{1}{4}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{1}{4} for x_{2}.
4x^{2}-7x-2=4\left(x-2\right)\left(x+\frac{1}{4}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
4x^{2}-7x-2=4\left(x-2\right)\times \left(\frac{4x+1}{4}\right)
Add \frac{1}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}-7x-2=\left(x-2\right)\left(4x+1\right)
Cancel out 4, the greatest common factor in 4 and 4.
x ^ 2 -\frac{7}{4}x -\frac{1}{2} = 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 4
r + s = \frac{7}{4} rs = -\frac{1}{2}
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{7}{8} - u s = \frac{7}{8} + u
Two numbers r and s sum up to \frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{4} = \frac{7}{8}. 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{7}{8} - u) (\frac{7}{8} + u) = -\frac{1}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{2}
\frac{49}{64} - u^2 = -\frac{1}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{1}{2}-\frac{49}{64} = -\frac{81}{64}
Simplify the expression by subtracting \frac{49}{64} on both sides
u^2 = \frac{81}{64} u = \pm\sqrt{\frac{81}{64}} = \pm \frac{9}{8}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{7}{8} - \frac{9}{8} = -0.250 s = \frac{7}{8} + \frac{9}{8} = 2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.