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

Similar Problems from Web Search

Share

a+b=12 ab=1\times 20=20
Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+20. 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(y^{2}+2y\right)+\left(10y+20\right)
Rewrite y^{2}+12y+20 as \left(y^{2}+2y\right)+\left(10y+20\right).
y\left(y+2\right)+10\left(y+2\right)
Factor out y in the first and 10 in the second group.
\left(y+2\right)\left(y+10\right)
Factor out common term y+2 by using distributive property.
y^{2}+12y+20=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.
y=\frac{-12±\sqrt{12^{2}-4\times 20}}{2}
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.
y=\frac{-12±\sqrt{144-4\times 20}}{2}
Square 12.
y=\frac{-12±\sqrt{144-80}}{2}
Multiply -4 times 20.
y=\frac{-12±\sqrt{64}}{2}
Add 144 to -80.
y=\frac{-12±8}{2}
Take the square root of 64.
y=-\frac{4}{2}
Now solve the equation y=\frac{-12±8}{2} when ± is plus. Add -12 to 8.
y=-2
Divide -4 by 2.
y=-\frac{20}{2}
Now solve the equation y=\frac{-12±8}{2} when ± is minus. Subtract 8 from -12.
y=-10
Divide -20 by 2.
y^{2}+12y+20=\left(y-\left(-2\right)\right)\left(y-\left(-10\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 -10 for x_{2}.
y^{2}+12y+20=\left(y+2\right)\left(y+10\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +12x +20 = 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.
r + s = -12 rs = 20
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 = -6 - u s = -6 + u
Two numbers r and s sum up to -12 exactly when the average of the two numbers is \frac{1}{2}*-12 = -6. 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>
(-6 - u) (-6 + u) = 20
To solve for unknown quantity u, substitute these in the product equation rs = 20
36 - u^2 = 20
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 20-36 = -16
Simplify the expression by subtracting 36 on both sides
u^2 = 16 u = \pm\sqrt{16} = \pm 4
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-6 - 4 = -10 s = -6 + 4 = -2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.