Solve w^2-25w+46

$$w ^ { 2 } - 25 w + 46$$

Factor

$$\left(w-23\right)\left(w-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as w^{2}+aw+bw+46. To find a and b, set up a system to be solved.

$$a+b=-25 ab=1\times 46=46$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 46.

$$-1,-46 -2,-23$$

Calculate the sum for each pair.

$$-1-46=-47 -2-23=-25$$

The solution is the pair that gives sum -25.

$$a=-23 b=-2$$

Rewrite w^{2}-25w+46 as \left(w^{2}-23w\right)+\left(-2w+46\right).

$$\left(w^{2}-23w\right)+\left(-2w+46\right)$$

Factor out w in the first and -2 in the second group.

$$w\left(w-23\right)-2\left(w-23\right)$$

Factor out common term w-23 by using distributive property.

$$\left(w-23\right)\left(w-2\right)$$

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.

$$w^{2}-25w+46=0$$

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.

$$w=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 46}}{2}$$

Square -25.

$$w=\frac{-\left(-25\right)±\sqrt{625-4\times 46}}{2}$$

Multiply -4 times 46.

$$w=\frac{-\left(-25\right)±\sqrt{625-184}}{2}$$

Add 625 to -184.

$$w=\frac{-\left(-25\right)±\sqrt{441}}{2}$$

Take the square root of 441.

$$w=\frac{-\left(-25\right)±21}{2}$$

The opposite of -25 is 25.

$$w=\frac{25±21}{2}$$

Now solve the equation w=\frac{25±21}{2} when ± is plus. Add 25 to 21.

$$w=\frac{46}{2}$$

Divide 46 by 2.

$$w=23$$

Now solve the equation w=\frac{25±21}{2} when ± is minus. Subtract 21 from 25.

$$w=\frac{4}{2}$$

Divide 4 by 2.

$$w=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 23 for x_{1} and 2 for x_{2}.

$$w^{2}-25w+46=\left(w-23\right)\left(w-2\right)$$

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.

$$x ^ 2 -25x +46 = 0$$

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

$$r + s = 25 rs = 46$$

Two numbers r and s sum up to 25 exactly when the average of the two numbers is \frac{1}{2}*25 = \frac{25}{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>

$$r = \frac{25}{2} - u s = \frac{25}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 46

$$(\frac{25}{2} - u) (\frac{25}{2} + u) = 46$$

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

$$\frac{625}{4} - u^2 = 46$$

Simplify the expression by subtracting \frac{625}{4} on both sides

$$-u^2 = 46-\frac{625}{4} = -\frac{441}{4}$$

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

$$u^2 = \frac{441}{4} u = \pm\sqrt{\frac{441}{4}} = \pm \frac{21}{2}$$

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

$$r =\frac{25}{2} - \frac{21}{2} = 2 s = \frac{25}{2} + \frac{21}{2} = 23$$

Evaluate

$$\left(w-23\right)\left(w-2\right)$$