Solve x^2-21x+98

$$x ^ { 2 } - 21 x + 98$$

Factor

$$\left(x-14\right)\left(x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+98. To find a and b, set up a system to be solved.

$$a+b=-21 ab=1\times 98=98$$

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 98.

$$-1,-98 -2,-49 -7,-14$$

Calculate the sum for each pair.

$$-1-98=-99 -2-49=-51 -7-14=-21$$

The solution is the pair that gives sum -21.

$$a=-14 b=-7$$

Rewrite x^{2}-21x+98 as \left(x^{2}-14x\right)+\left(-7x+98\right).

$$\left(x^{2}-14x\right)+\left(-7x+98\right)$$

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

$$x\left(x-14\right)-7\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x-7\right)$$

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.

$$x^{2}-21x+98=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 98}}{2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 98}}{2}$$

Multiply -4 times 98.

$$x=\frac{-\left(-21\right)±\sqrt{441-392}}{2}$$

Add 441 to -392.

$$x=\frac{-\left(-21\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-\left(-21\right)±7}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±7}{2}$$

Now solve the equation x=\frac{21±7}{2} when ± is plus. Add 21 to 7.

$$x=\frac{28}{2}$$

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{21±7}{2} when ± is minus. Subtract 7 from 21.

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

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 14 for x_{1} and 7 for x_{2}.

$$x^{2}-21x+98=\left(x-14\right)\left(x-7\right)$$

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.

$$x ^ 2 -21x +98 = 0$$

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

$$r + s = 21 rs = 98$$

Two numbers r and s sum up to 21 exactly when the average of the two numbers is \frac{1}{2}*21 = \frac{21}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = 98

$$(\frac{21}{2} - u) (\frac{21}{2} + u) = 98$$

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

$$\frac{441}{4} - u^2 = 98$$

Simplify the expression by subtracting \frac{441}{4} on both sides

$$-u^2 = 98-\frac{441}{4} = -\frac{49}{4}$$

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

$$u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}$$

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

$$r =\frac{21}{2} - \frac{7}{2} = 7 s = \frac{21}{2} + \frac{7}{2} = 14$$

Evaluate

$$\left(x-14\right)\left(x-7\right)$$