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

Similar Problems from Web Search

Share

a+b=7 ab=1\times 6=6
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+6. To find a and b, set up a system to be solved.
1,6 2,3
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 6.
1+6=7 2+3=5
Calculate the sum for each pair.
a=1 b=6
The solution is the pair that gives sum 7.
\left(x^{2}+x\right)+\left(6x+6\right)
Rewrite x^{2}+7x+6 as \left(x^{2}+x\right)+\left(6x+6\right).
x\left(x+1\right)+6\left(x+1\right)
Factor out x in the first and 6 in the second group.
\left(x+1\right)\left(x+6\right)
Factor out common term x+1 by using distributive property.
x^{2}+7x+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.
x=\frac{-7±\sqrt{7^{2}-4\times 6}}{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.
x=\frac{-7±\sqrt{49-4\times 6}}{2}
Square 7.
x=\frac{-7±\sqrt{49-24}}{2}
Multiply -4 times 6.
x=\frac{-7±\sqrt{25}}{2}
Add 49 to -24.
x=\frac{-7±5}{2}
Take the square root of 25.
x=-\frac{2}{2}
Now solve the equation x=\frac{-7±5}{2} when ± is plus. Add -7 to 5.
x=-1
Divide -2 by 2.
x=-\frac{12}{2}
Now solve the equation x=\frac{-7±5}{2} when ± is minus. Subtract 5 from -7.
x=-6
Divide -12 by 2.
x^{2}+7x+6=\left(x-\left(-1\right)\right)\left(x-\left(-6\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -6 for x_{2}.
x^{2}+7x+6=\left(x+1\right)\left(x+6\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +7x +6 = 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 = -7 rs = 6
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}{2} - u s = -\frac{7}{2} + u
Two numbers r and s sum up to -7 exactly when the average of the two numbers is \frac{1}{2}*-7 = -\frac{7}{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{7}{2} - u) (-\frac{7}{2} + u) = 6
To solve for unknown quantity u, substitute these in the product equation rs = 6
\frac{49}{4} - u^2 = 6
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 6-\frac{49}{4} = -\frac{25}{4}
Simplify the expression by subtracting \frac{49}{4} on both sides
u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{2}
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}{2} - \frac{5}{2} = -6 s = -\frac{7}{2} + \frac{5}{2} = -1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.