Solve x^2-16x+63

$$x ^ { 2 } - 16 x + 63$$

Factor

$$\left(x-9\right)\left(x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+63. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 63=63$$

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 63.

$$-1,-63 -3,-21 -7,-9$$

Calculate the sum for each pair.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

The solution is the pair that gives sum -16.

$$a=-9 b=-7$$

Rewrite x^{2}-16x+63 as \left(x^{2}-9x\right)+\left(-7x+63\right).

$$\left(x^{2}-9x\right)+\left(-7x+63\right)$$

Factor out x in the first and -7 in the second group.

$$x\left(x-9\right)-7\left(x-9\right)$$

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

$$\left(x-9\right)\left(x-7\right)$$

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.

$$x^{2}-16x+63=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 63}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 63}}{2}$$

Multiply -4 times 63.

$$x=\frac{-\left(-16\right)±\sqrt{256-252}}{2}$$

Add 256 to -252.

$$x=\frac{-\left(-16\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-\left(-16\right)±2}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±2}{2}$$

Now solve the equation x=\frac{16±2}{2} when ± is plus. Add 16 to 2.

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

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{16±2}{2} when ± is minus. Subtract 2 from 16.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and 7 for x_{2}.

$$x^{2}-16x+63=\left(x-9\right)\left(x-7\right)$$

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.

$$x ^ 2 -16x +63 = 0$$

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

$$r + s = 16 rs = 63$$

Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>

$$r = 8 - u s = 8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 63

$$(8 - u) (8 + u) = 63$$

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

$$64 - u^2 = 63$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 63-64 = -1$$

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

$$u^2 = 1 u = \pm\sqrt{1} = \pm 1$$

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

$$r =8 - 1 = 7 s = 8 + 1 = 9$$

Evaluate

$$\left(x-9\right)\left(x-7\right)$$