Solve B^2+6B+8 | Microsoft Math Solver (2024)

a+b=6 ab=1\times 8=8

Factor the expression by grouping. First, the expression needs to be rewritten as B^{2}+aB+bB+8. To find a and b, set up a system to be solved.

1,8 2,4

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 8.

1+8=9 2+4=6

Calculate the sum for each pair.

a=2 b=4

The solution is the pair that gives sum 6.

\left(B^{2}+2B\right)+\left(4B+8\right)

Rewrite B^{2}+6B+8 as \left(B^{2}+2B\right)+\left(4B+8\right).

B\left(B+2\right)+4\left(B+2\right)

Factor out B in the first and 4 in the second group.

\left(B+2\right)\left(B+4\right)

Factor out common term B+2 by using distributive property.

B^{2}+6B+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.

B=\frac{-6±\sqrt{6^{2}-4\times 8}}{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.

B=\frac{-6±\sqrt{36-4\times 8}}{2}

Square 6.

B=\frac{-6±\sqrt{36-32}}{2}

Multiply -4 times 8.

B=\frac{-6±\sqrt{4}}{2}

Add 36 to -32.

B=\frac{-6±2}{2}

Take the square root of 4.

B=-\frac{4}{2}

Now solve the equation B=\frac{-6±2}{2} when ± is plus. Add -6 to 2.

B=-2

Divide -4 by 2.

B=-\frac{8}{2}

Now solve the equation B=\frac{-6±2}{2} when ± is minus. Subtract 2 from -6.

B=-4

Divide -8 by 2.

B^{2}+6B+8=\left(B-\left(-2\right)\right)\left(B-\left(-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 -4 for x_{2}.

B^{2}+6B+8=\left(B+2\right)\left(B+4\right)

Simplify all the expressions of the form p-\left(-q\right) to p+q.

x ^ 2 +6x +8 = 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 = -6 rs = 8

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 = -3 - u s = -3 + u

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>

(-3 - u) (-3 + u) = 8

To solve for unknown quantity u, substitute these in the product equation rs = 8

9 - u^2 = 8

Simplify by expanding (a -b) (a + b) = a^2 – b^2

-u^2 = 8-9 = -1

Simplify the expression by subtracting 9 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 =-3 - 1 = -4 s = -3 + 1 = -2

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.