Solve -x^2+14x-45

$$- x ^ { 2 } + 14 x - 45$$

Factor

$$\left(5-x\right)\left(x-9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-45. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-45\right)=45$$

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 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 14.

$$a=9 b=5$$

Rewrite -x^{2}+14x-45 as \left(-x^{2}+9x\right)+\left(5x-45\right).

$$\left(-x^{2}+9x\right)+\left(5x-45\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-9\right)+5\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(-x+5\right)$$

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.

$$-x^{2}+14x-45=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-45\right)}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-45\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\left(-45\right)}}{2\left(-1\right)}$$

Multiply 4 times -45.

$$x=\frac{-14±\sqrt{196-180}}{2\left(-1\right)}$$

Add 196 to -180.

$$x=\frac{-14±\sqrt{16}}{2\left(-1\right)}$$

Take the square root of 16.

$$x=\frac{-14±4}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14±4}{-2}$$

Now solve the equation x=\frac{-14±4}{-2} when ± is plus. Add -14 to 4.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-14±4}{-2} when ± is minus. Subtract 4 from -14.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and 9 for x_{2}.

$$-x^{2}+14x-45=-\left(x-5\right)\left(x-9\right)$$

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.

$$x ^ 2 -14x +45 = 0$$

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

$$r + s = 14 rs = 45$$

Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>

$$r = 7 - u s = 7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 45

$$(7 - u) (7 + u) = 45$$

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

$$49 - u^2 = 45$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 45-49 = -4$$

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

$$u^2 = 4 u = \pm\sqrt{4} = \pm 2$$

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

$$r =7 - 2 = 5 s = 7 + 2 = 9$$

Evaluate

$$\left(5-x\right)\left(x-9\right)$$