Solve 20a^2-7a-40

$$20 a ^ { 2 } - 7 a - 40$$

Factor

$$\left(5a-8\right)\left(4a+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 20a^{2}+pa+qa-40. To find p and q, set up a system to be solved.

$$p+q=-7 pq=20\left(-40\right)=-800$$

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 -800.

$$1,-800 2,-400 4,-200 5,-160 8,-100 10,-80 16,-50 20,-40 25,-32$$

Calculate the sum for each pair.

$$1-800=-799 2-400=-398 4-200=-196 5-160=-155 8-100=-92 10-80=-70 16-50=-34 20-40=-20 25-32=-7$$

The solution is the pair that gives sum -7.

$$p=-32 q=25$$

Rewrite 20a^{2}-7a-40 as \left(20a^{2}-32a\right)+\left(25a-40\right).

$$\left(20a^{2}-32a\right)+\left(25a-40\right)$$

Factor out 4a in the first and 5 in the second group.

$$4a\left(5a-8\right)+5\left(5a-8\right)$$

Factor out common term 5a-8 by using distributive property.

$$\left(5a-8\right)\left(4a+5\right)$$

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.

$$20a^{2}-7a-40=0$$

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.

$$a=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 20\left(-40\right)}}{2\times 20}$$

Square -7.

$$a=\frac{-\left(-7\right)±\sqrt{49-4\times 20\left(-40\right)}}{2\times 20}$$

Multiply -4 times 20.

$$a=\frac{-\left(-7\right)±\sqrt{49-80\left(-40\right)}}{2\times 20}$$

Multiply -80 times -40.

$$a=\frac{-\left(-7\right)±\sqrt{49+3200}}{2\times 20}$$

Add 49 to 3200.

$$a=\frac{-\left(-7\right)±\sqrt{3249}}{2\times 20}$$

Take the square root of 3249.

$$a=\frac{-\left(-7\right)±57}{2\times 20}$$

The opposite of -7 is 7.

$$a=\frac{7±57}{2\times 20}$$

Multiply 2 times 20.

$$a=\frac{7±57}{40}$$

Now solve the equation a=\frac{7±57}{40} when ± is plus. Add 7 to 57.

$$a=\frac{64}{40}$$

Reduce the fraction \frac{64}{40} to lowest terms by extracting and canceling out 8.

$$a=\frac{8}{5}$$

Now solve the equation a=\frac{7±57}{40} when ± is minus. Subtract 57 from 7.

$$a=-\frac{50}{40}$$

Reduce the fraction \frac{-50}{40} to lowest terms by extracting and canceling out 10.

$$a=-\frac{5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{8}{5} for x_{1} and -\frac{5}{4} for x_{2}.

$$20a^{2}-7a-40=20\left(a-\frac{8}{5}\right)\left(a-\left(-\frac{5}{4}\right)\right)$$

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

$$20a^{2}-7a-40=20\left(a-\frac{8}{5}\right)\left(a+\frac{5}{4}\right)$$

Subtract \frac{8}{5} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$20a^{2}-7a-40=20\times \frac{5a-8}{5}\left(a+\frac{5}{4}\right)$$

Add \frac{5}{4} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$20a^{2}-7a-40=20\times \frac{5a-8}{5}\times \frac{4a+5}{4}$$

Multiply \frac{5a-8}{5} times \frac{4a+5}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$20a^{2}-7a-40=20\times \frac{\left(5a-8\right)\left(4a+5\right)}{5\times 4}$$

Multiply 5 times 4.

$$20a^{2}-7a-40=20\times \frac{\left(5a-8\right)\left(4a+5\right)}{20}$$

Cancel out 20, the greatest common factor in 20 and 20.

$$20a^{2}-7a-40=\left(5a-8\right)\left(4a+5\right)$$

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 20

$$x ^ 2 -\frac{7}{20}x -2 = 0$$

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

$$r + s = \frac{7}{20} rs = -2$$

Two numbers r and s sum up to \frac{7}{20} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{20} = \frac{7}{40}. 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>

$$r = \frac{7}{40} - u s = \frac{7}{40} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

$$(\frac{7}{40} - u) (\frac{7}{40} + u) = -2$$

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

$$\frac{49}{1600} - u^2 = -2$$

Simplify the expression by subtracting \frac{49}{1600} on both sides

$$-u^2 = -2-\frac{49}{1600} = -\frac{3249}{1600}$$

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

$$u^2 = \frac{3249}{1600} u = \pm\sqrt{\frac{3249}{1600}} = \pm \frac{57}{40}$$

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

$$r =\frac{7}{40} - \frac{57}{40} = -1.250 s = \frac{7}{40} + \frac{57}{40} = 1.600$$

Evaluate

$$\left(5a-8\right)\left(4a+5\right)$$