Solve 2a^2-7a-4

$$2 a ^ { 2 } - 7 a - 4$$

Factor

$$\left(a-4\right)\left(2a+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2a^{2}+pa+qa-4. To find p and q, set up a system to be solved.

$$p+q=-7 pq=2\left(-4\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -7.

$$p=-8 q=1$$

Rewrite 2a^{2}-7a-4 as \left(2a^{2}-8a\right)+\left(a-4\right).

$$\left(2a^{2}-8a\right)+\left(a-4\right)$$

Factor out 2a in 2a^{2}-8a.

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

Factor out common term a-4 by using distributive property.

$$\left(a-4\right)\left(2a+1\right)$$

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.

$$2a^{2}-7a-4=0$$

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.

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

Square -7.

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

Multiply -4 times 2.

$$a=\frac{-\left(-7\right)±\sqrt{49-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$a=\frac{-\left(-7\right)±\sqrt{49+32}}{2\times 2}$$

Add 49 to 32.

$$a=\frac{-\left(-7\right)±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$a=\frac{-\left(-7\right)±9}{2\times 2}$$

The opposite of -7 is 7.

$$a=\frac{7±9}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{7±9}{4}$$

Now solve the equation a=\frac{7±9}{4} when ± is plus. Add 7 to 9.

$$a=\frac{16}{4}$$

Divide 16 by 4.

$$a=4$$

Now solve the equation a=\frac{7±9}{4} when ± is minus. Subtract 9 from 7.

$$a=-\frac{2}{4}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$a=-\frac{1}{2}$$

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

$$2a^{2}-7a-4=2\left(a-4\right)\left(a-\left(-\frac{1}{2}\right)\right)$$

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

$$2a^{2}-7a-4=2\left(a-4\right)\left(a+\frac{1}{2}\right)$$

Add \frac{1}{2} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2a^{2}-7a-4=2\left(a-4\right)\times \frac{2a+1}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2a^{2}-7a-4=\left(a-4\right)\left(2a+1\right)$$

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 2

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

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

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

Two numbers r and s sum up to \frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{2} = \frac{7}{4}. 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>

$$r = \frac{7}{4} - u s = \frac{7}{4} + u$$

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

$$(\frac{7}{4} - u) (\frac{7}{4} + u) = -2$$

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

$$\frac{49}{16} - u^2 = -2$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = -2-\frac{49}{16} = -\frac{81}{16}$$

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

$$u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}$$

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

$$r =\frac{7}{4} - \frac{9}{4} = -0.500 s = \frac{7}{4} + \frac{9}{4} = 4$$

Evaluate

$$\left(a-4\right)\left(2a+1\right)$$