Solve 2g^2+44g-46

$$2 g ^ { 2 } + 44 g - 46$$

Factor

$$2\left(g-1\right)\left(g+23\right)$$

Solution Steps

Factor out 2.

$$2\left(g^{2}+22g-23\right)$$

Consider g^{2}+22g-23. Factor the expression by grouping. First, the expression needs to be rewritten as g^{2}+ag+bg-23. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\left(-23\right)=-23$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=23$$

Rewrite g^{2}+22g-23 as \left(g^{2}-g\right)+\left(23g-23\right).

$$\left(g^{2}-g\right)+\left(23g-23\right)$$

Factor out g in the first and 23 in the second group.

$$g\left(g-1\right)+23\left(g-1\right)$$

Factor out common term g-1 by using distributive property.

$$\left(g-1\right)\left(g+23\right)$$

Rewrite the complete factored expression.

$$2\left(g-1\right)\left(g+23\right)$$

Steps Using the Quadratic Formula

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.

$$2g^{2}+44g-46=0$$

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.

$$g=\frac{-44±\sqrt{44^{2}-4\times 2\left(-46\right)}}{2\times 2}$$

Square 44.

$$g=\frac{-44±\sqrt{1936-4\times 2\left(-46\right)}}{2\times 2}$$

Multiply -4 times 2.

$$g=\frac{-44±\sqrt{1936-8\left(-46\right)}}{2\times 2}$$

Multiply -8 times -46.

$$g=\frac{-44±\sqrt{1936+368}}{2\times 2}$$

Add 1936 to 368.

$$g=\frac{-44±\sqrt{2304}}{2\times 2}$$

Take the square root of 2304.

$$g=\frac{-44±48}{2\times 2}$$

Multiply 2 times 2.

$$g=\frac{-44±48}{4}$$

Now solve the equation g=\frac{-44±48}{4} when ± is plus. Add -44 to 48.

$$g=\frac{4}{4}$$

Divide 4 by 4.

$$g=1$$

Now solve the equation g=\frac{-44±48}{4} when ± is minus. Subtract 48 from -44.

$$g=-\frac{92}{4}$$

Divide -92 by 4.

$$g=-23$$

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 -23 for x_{2}.

$$2g^{2}+44g-46=2\left(g-1\right)\left(g-\left(-23\right)\right)$$

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

$$2g^{2}+44g-46=2\left(g-1\right)\left(g+23\right)$$

Steps Using Direct Factoring Method

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 2

$$x ^ 2 +22x -23 = 0$$

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

$$r + s = -22 rs = -23$$

Two numbers r and s sum up to -22 exactly when the average of the two numbers is \frac{1}{2}*-22 = -11. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -11 - u s = -11 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -23

$$(-11 - u) (-11 + u) = -23$$

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

$$121 - u^2 = -23$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = -23-121 = -144$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 144 u = \pm\sqrt{144} = \pm 12$$

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

$$r =-11 - 12 = -23 s = -11 + 12 = 1$$

Evaluate

$$2\left(g-1\right)\left(g+23\right)$$