Solve 7a^2-14a-21

$$7 a ^ { 2 } - 14 a - 21$$

Factor

$$7\left(a-3\right)\left(a+1\right)$$

Solution Steps

Factor out 7.

$$7\left(a^{2}-2a-3\right)$$

Consider a^{2}-2a-3. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-3. To find p and q, set up a system to be solved.

$$p+q=-2 pq=1\left(-3\right)=-3$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$p=-3 q=1$$

Rewrite a^{2}-2a-3 as \left(a^{2}-3a\right)+\left(a-3\right).

$$\left(a^{2}-3a\right)+\left(a-3\right)$$

Factor out a in a^{2}-3a.

$$a\left(a-3\right)+a-3$$

Factor out common term a-3 by using distributive property.

$$\left(a-3\right)\left(a+1\right)$$

Rewrite the complete factored expression.

$$7\left(a-3\right)\left(a+1\right)$$

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.

$$7a^{2}-14a-21=0$$

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.

$$a=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 7\left(-21\right)}}{2\times 7}$$

Square -14.

$$a=\frac{-\left(-14\right)±\sqrt{196-4\times 7\left(-21\right)}}{2\times 7}$$

Multiply -4 times 7.

$$a=\frac{-\left(-14\right)±\sqrt{196-28\left(-21\right)}}{2\times 7}$$

Multiply -28 times -21.

$$a=\frac{-\left(-14\right)±\sqrt{196+588}}{2\times 7}$$

Add 196 to 588.

$$a=\frac{-\left(-14\right)±\sqrt{784}}{2\times 7}$$

Take the square root of 784.

$$a=\frac{-\left(-14\right)±28}{2\times 7}$$

The opposite of -14 is 14.

$$a=\frac{14±28}{2\times 7}$$

Multiply 2 times 7.

$$a=\frac{14±28}{14}$$

Now solve the equation a=\frac{14±28}{14} when ± is plus. Add 14 to 28.

$$a=\frac{42}{14}$$

Divide 42 by 14.

$$a=3$$

Now solve the equation a=\frac{14±28}{14} when ± is minus. Subtract 28 from 14.

$$a=-\frac{14}{14}$$

Divide -14 by 14.

$$a=-1$$

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 -1 for x_{2}.

$$7a^{2}-14a-21=7\left(a-3\right)\left(a-\left(-1\right)\right)$$

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

$$7a^{2}-14a-21=7\left(a-3\right)\left(a+1\right)$$

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.This is achieved by dividing both sides of the equation by 7

$$x ^ 2 -2x -3 = 0$$

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

$$r + s = 2 rs = -3$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -3

$$(1 - u) (1 + u) = -3$$

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

$$1 - u^2 = -3$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -3-1 = -4$$

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.

$$r =1 - 2 = -1 s = 1 + 2 = 3$$

Evaluate

$$7\left(a-3\right)\left(a+1\right)$$