Solve f(x)=x^2+10x+21

$$f ( x ) = x ^ { 2 } + 10 x + 21$$

Factor

$$\left(x+3\right)\left(x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=10 ab=1\times 21=21$$

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 21.

$$1,21 3,7$$

Calculate the sum for each pair.

$$1+21=22 3+7=10$$

The solution is the pair that gives sum 10.

$$a=3 b=7$$

Rewrite x^{2}+10x+21 as \left(x^{2}+3x\right)+\left(7x+21\right).

$$\left(x^{2}+3x\right)+\left(7x+21\right)$$

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

$$x\left(x+3\right)+7\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+7\right)$$

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.

$$x^{2}+10x+21=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 21}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 21}}{2}$$

Multiply -4 times 21.

$$x=\frac{-10±\sqrt{100-84}}{2}$$

Add 100 to -84.

$$x=\frac{-10±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-10±4}{2}$$

Now solve the equation x=\frac{-10±4}{2} when ± is plus. Add -10 to 4.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-10±4}{2} when ± is minus. Subtract 4 from -10.

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

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 -3 for x_{1} and -7 for x_{2}.

$$x^{2}+10x+21=\left(x-\left(-3\right)\right)\left(x-\left(-7\right)\right)$$

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

$$x^{2}+10x+21=\left(x+3\right)\left(x+7\right)$$

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.

$$x ^ 2 +10x +21 = 0$$

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

$$r + s = -10 rs = 21$$

Two numbers r and s sum up to -10 exactly when the average of the two numbers is \frac{1}{2}*-10 = -5. 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>

$$r = -5 - u s = -5 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 21

$$(-5 - u) (-5 + u) = 21$$

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

$$25 - u^2 = 21$$

Simplify the expression by subtracting 25 on both sides

$$-u^2 = 21-25 = -4$$

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.

$$r =-5 - 2 = -7 s = -5 + 2 = -3$$

Evaluate

$$\left(x+3\right)\left(x+7\right)$$