Solve 8a^2-22a-21

$$8 a ^ { 2 } - 22 a - 21$$

Factor

$$\left(2a-7\right)\left(4a+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8a^{2}+pa+qa-21. To find p and q, set up a system to be solved.

$$p+q=-22 pq=8\left(-21\right)=-168$$

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. List all such integer pairs that give product -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -22.

$$p=-28 q=6$$

Rewrite 8a^{2}-22a-21 as \left(8a^{2}-28a\right)+\left(6a-21\right).

$$\left(8a^{2}-28a\right)+\left(6a-21\right)$$

Factor out 4a in the first and 3 in the second group.

$$4a\left(2a-7\right)+3\left(2a-7\right)$$

Factor out common term 2a-7 by using distributive property.

$$\left(2a-7\right)\left(4a+3\right)$$

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.

$$8a^{2}-22a-21=0$$

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.

$$a=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\left(-21\right)}}{2\times 8}$$

Square -22.

$$a=\frac{-\left(-22\right)±\sqrt{484-4\times 8\left(-21\right)}}{2\times 8}$$

Multiply -4 times 8.

$$a=\frac{-\left(-22\right)±\sqrt{484-32\left(-21\right)}}{2\times 8}$$

Multiply -32 times -21.

$$a=\frac{-\left(-22\right)±\sqrt{484+672}}{2\times 8}$$

Add 484 to 672.

$$a=\frac{-\left(-22\right)±\sqrt{1156}}{2\times 8}$$

Take the square root of 1156.

$$a=\frac{-\left(-22\right)±34}{2\times 8}$$

The opposite of -22 is 22.

$$a=\frac{22±34}{2\times 8}$$

Multiply 2 times 8.

$$a=\frac{22±34}{16}$$

Now solve the equation a=\frac{22±34}{16} when ± is plus. Add 22 to 34.

$$a=\frac{56}{16}$$

Reduce the fraction \frac{56}{16} to lowest terms by extracting and canceling out 8.

$$a=\frac{7}{2}$$

Now solve the equation a=\frac{22±34}{16} when ± is minus. Subtract 34 from 22.

$$a=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

$$a=-\frac{3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and -\frac{3}{4} for x_{2}.

$$8a^{2}-22a-21=8\left(a-\frac{7}{2}\right)\left(a-\left(-\frac{3}{4}\right)\right)$$

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

$$8a^{2}-22a-21=8\left(a-\frac{7}{2}\right)\left(a+\frac{3}{4}\right)$$

Subtract \frac{7}{2} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8a^{2}-22a-21=8\times \frac{2a-7}{2}\left(a+\frac{3}{4}\right)$$

Add \frac{3}{4} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8a^{2}-22a-21=8\times \frac{2a-7}{2}\times \frac{4a+3}{4}$$

Multiply \frac{2a-7}{2} times \frac{4a+3}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8a^{2}-22a-21=8\times \frac{\left(2a-7\right)\left(4a+3\right)}{2\times 4}$$

Multiply 2 times 4.

$$8a^{2}-22a-21=8\times \frac{\left(2a-7\right)\left(4a+3\right)}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8a^{2}-22a-21=\left(2a-7\right)\left(4a+3\right)$$

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 8

$$x ^ 2 -\frac{11}{4}x -\frac{21}{8} = 0$$

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

$$r + s = \frac{11}{4} rs = -\frac{21}{8}$$

Two numbers r and s sum up to \frac{11}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{4} = \frac{11}{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>

$$r = \frac{11}{8} - u s = \frac{11}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{8}

$$(\frac{11}{8} - u) (\frac{11}{8} + u) = -\frac{21}{8}$$

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

$$\frac{121}{64} - u^2 = -\frac{21}{8}$$

Simplify the expression by subtracting \frac{121}{64} on both sides

$$-u^2 = -\frac{21}{8}-\frac{121}{64} = -\frac{289}{64}$$

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

$$u^2 = \frac{289}{64} u = \pm\sqrt{\frac{289}{64}} = \pm \frac{17}{8}$$

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

$$r =\frac{11}{8} - \frac{17}{8} = -0.750 s = \frac{11}{8} + \frac{17}{8} = 3.500$$

Evaluate

$$\left(2a-7\right)\left(4a+3\right)$$