Factor
\left(2x+1\right)\left(2x+5\right)
Evaluate
\left(2x+1\right)\left(2x+5\right)
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a+b=12 ab=4\times 5=20
Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+5. To find a and b, set up a system to be solved.
1,20 2,10 4,5
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 20.
1+20=21 2+10=12 4+5=9
Calculate the sum for each pair.
a=2 b=10
The solution is the pair that gives sum 12.
\left(4x^{2}+2x\right)+\left(10x+5\right)
Rewrite 4x^{2}+12x+5 as \left(4x^{2}+2x\right)+\left(10x+5\right).
2x\left(2x+1\right)+5\left(2x+1\right)
Factor out 2x in the first and 5 in the second group.
\left(2x+1\right)\left(2x+5\right)
Factor out common term 2x+1 by using distributive property.
4x^{2}+12x+5=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{-12±\sqrt{12^{2}-4\times 4\times 5}}{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{-12±\sqrt{144-4\times 4\times 5}}{2\times 4}
Square 12.
x=\frac{-12±\sqrt{144-16\times 5}}{2\times 4}
Multiply -4 times 4.
x=\frac{-12±\sqrt{144-80}}{2\times 4}
Multiply -16 times 5.
x=\frac{-12±\sqrt{64}}{2\times 4}
Add 144 to -80.
x=\frac{-12±8}{2\times 4}
Take the square root of 64.
x=\frac{-12±8}{8}
Multiply 2 times 4.
x=-\frac{4}{8}
Now solve the equation x=\frac{-12±8}{8} when ± is plus. Add -12 to 8.
x=-\frac{1}{2}
Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.
x=-\frac{20}{8}
Now solve the equation x=\frac{-12±8}{8} when ± is minus. Subtract 8 from -12.
x=-\frac{5}{2}
Reduce the fraction \frac{-20}{8} to lowest terms by extracting and canceling out 4.
4x^{2}+12x+5=4\left(x-\left(-\frac{1}{2}\right)\right)\left(x-\left(-\frac{5}{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{1}{2} for x_{1} and -\frac{5}{2} for x_{2}.
4x^{2}+12x+5=4\left(x+\frac{1}{2}\right)\left(x+\frac{5}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
4x^{2}+12x+5=4\times \frac{2x+1}{2}\left(x+\frac{5}{2}\right)
Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}+12x+5=4\times \frac{2x+1}{2}\times \frac{2x+5}{2}
Add \frac{5}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}+12x+5=4\times \frac{\left(2x+1\right)\left(2x+5\right)}{2\times 2}
Multiply \frac{2x+1}{2} times \frac{2x+5}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
4x^{2}+12x+5=4\times \frac{\left(2x+1\right)\left(2x+5\right)}{4}
Multiply 2 times 2.
4x^{2}+12x+5=\left(2x+1\right)\left(2x+5\right)
Cancel out 4, the greatest common factor in 4 and 4.
x ^ 2 +3x +\frac{5}{4} = 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 = -3 rs = \frac{5}{4}
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{3}{2} - u s = -\frac{3}{2} + u
Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}*-3 = -\frac{3}{2}. 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{3}{2} - u) (-\frac{3}{2} + u) = \frac{5}{4}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{4}
\frac{9}{4} - u^2 = \frac{5}{4}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{5}{4}-\frac{9}{4} = -1
Simplify the expression by subtracting \frac{9}{4} on both sides
u^2 = 1 u = \pm\sqrt{1} = \pm 1
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{3}{2} - 1 = -2.500 s = -\frac{3}{2} + 1 = -0.500
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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\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
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